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0, we can assume that p(x) = 1. Choose a subsequence {X n (i)} of {Xn } such that I X n (i) 1 ...... 1 . By taking a further subsequence we can assume that the X n (i) have the same sign and, since p(x) = p( -x), it suffices to consider the case X n(i) > 0 for all i. Define Yn = 0 if either n i= n( i) or n = n( i) with i odd, while Yn = 1 if n = n(i) with i even. Then
c1 (P ( X + t y) - p( x)]
=
1 if t > 0, = 0 if - 1 < t < O.
Despite the foregoing example, there are nonseparable spaces in which the conclusion to Mazur's theorem remains valid, for instance, the class of weakly compactly generated (WCG) Banach spaces (to which we will return later). The attempt to characterize those spaces in which convex continuous functions are always generically differentiable has motivated the following terminology.
Definition 1.22. A Banach space E is said to be an Asplund space ( weak As plund space) provided every continuous convex function defined on a nonempty open convex subset D of E is Frechet differentiable (Gateaux differentiable) at each point of some dense G6 subset of D. Note that the term "weak" does not refer to the weak topology; it arose because Gateaux differentiability has sometimes been called "weak differentiability" (since it is weaker than Fnkhet - sometimes called "strong"- differentiability).
Some 30 years after Mazur's paper, J. Lindenstrauss [Li] obtained the next result of this nature, when he showed that if a separable Banach space was also reflexive, then the Gateaux differentiability in Mazur's theorem could be changed from Gateaux to Frechet. Five years later, E. Asplund renewed the study of such questions [Asp], making some impressive contributions. Mazur's theorem can be restated in the form "Separable Banach spaces are weak As plund spaces" (while Example 1.21 shows that £00 is not a weak Asplund space and Example 1. 14(a) shows that £1 is not an Asplund space). We will even tually prove, among other things, the fundamental result (due to Asplund) which states that if E* is separable, then E is an Asplund space ; this clearly implies Lindenstrauss' theorem.
14
Section 1
For convex continuous functions f, it is possible (and quite useful) to characterize Frechet differentiability at x solely in terms of f, that is, without mentioning the linear functional f' (x). (This is possible since there are always candidates for the derivative lurking in af(x) . )
Proposition 1 .23. Suppose that f is continuous and convex on the nonempty open convex subset D of E. Then f is Frechet differentiable at x E D if and only if for all f > 0 there exists 6 > 0 such that f(x + ty) + f(x - ty) - 2f(x) whenever lI y ll
=
1 and 0
0 there exists 6 > 0 such that (1. 7) holds for any o < t < 6 and any y with lI y ll = 1, that is,
f(x + t y) - f(x) - (x * , ty) ::; tf + f(x) - f(x - t y) - (x * , ty) . Inequality ( 1. 8) shows that the left side is greater or equal to 0 while ( 1.9) shows that the right side is less than or equal to tf, which completes the proof.
1.24. Exercise. Prove that a continuous convex function f on the open convex set D is Gateaux differentiable at xED if and only if for each yE E
t ) t ) lim t_o+ f(x + y + f(x - y t
-
2f(x)
=
.
O
The previous proposition makes it easy to formulate and prove the follow ing basic fact.
Proposition 1.25. Suppose that f is continuous and convex on the nonempty open convex subset D of E. Then the set G (possibly empty) of po ints x E D where f is Frechet differentiable is a G6. Proof. For each n let Gn be the set of all those x E D for which there exists > 0 such that
6
1. Convex functions on real Banach spaces.
sup
f(x + 6y) + f(x - 6y) - 2f(x) 6
0 and M > 0 such that If(u) - f( v)1 ::; Mllu - vII whenever u, v E B(Xj 6 ), Since x E Gn, there exist 1 6> 0 and r> 0 such that for all y with Ilyll = 1 we have x ± 6y E D and
f(x + 6y) + f(x - 6y) - 2f(x) < 6
r
'( ,x ) - 4>'( ,x o) = =
(df( h + ( 1 - ,xy) , x - y) - f(x) + fey) - (df( ,x ox + ( 1 - ,xOy), x - y) + f(x) - fey) = (df(,xx + ( 1 - ,x ) y) - df(,xox + ( 1 - ,xo )y), x - y) .
Now, x -y = (A !A O ) { [,xx + ( I - ,x ) y] - [ ,x ox + ( 1 - ,x o) y] } , so by monotonicity of df (and the fact that p !A O ) > 0), we have 4>' ( ,x ) � 4>'( ,x o ) = 0, that is, 4> is nondecreasing on ( ,x o , 1 ], a contradiction. (e) This example arises in fixed-point theory. Let C be a bounded closed convex nonempty subset of Hilbert space H and let U be a (generally nonlinear) nonexpan sive map of C into itself: I I U(x) - U ( Y ) II :5 I l x - yll for all x, y E C. Let I denote the identity map in Hj then T = I - U is monotone, with D(T) = C. Indeed, for all x , y E C, ( T(x) - T( y), x - y) = (x - y - ( U (x) - U( y)), x - y) = 2 2 = I l x - Yl1 - ( U(x) - U( y), x - y) � I l x - Y l 1 - 11U (x) - U( y) II ' ll x - yll � O. Note that 0 is in the range of T if and only if U has a fixed point in C; this hints at the importance for applications of studying the ranges of monotone operators (something we will not do, but which is done, for instance, in [Au-Ek], [pa-Sb] and [ZeD · (f) Again, in Hilbert space, let C be a nonempty closed convex set and let P be the metric projection of H onto C defined in Example 1 .14 (d). Since (as shown there) P is the Frechet derivative of a convex continuous function on H, it is monotone. More directly, this is an immediate consequence of inequality ( 1 .6).
Definition 2.3. Let X, Y be Hausdorff topological spaces and suppose that T: X -t 2 Y is a set-valued mapping from X into the subsets of Y. If A is a subset of E , we define T(A) = U{T(x): x E A } . We say that T is upper semicontinuous at the point x E X if, for each open set V in Y containing T(x), there is an open neighborhood U of x such that T(U) C V . Upper semicontinuity on a set is defined in the obvious way. 2.4. Exercises.
(a) Prove that a set-valued map T: E -+ 2 E * is norm-to-weak* [norm-to-norm] upper semicontinuous at x E E if and only iffor every weak* open set [norm open set] V containing T(x) and every {xn} C D(T) with I Ixn - x I I -+ 0, we have T(xn ) C V for all sufficiently large n. (Equivalently, T[B(x ; o)] C V for all sufficiently small 0 > 0.) (b) Assuming that T(x) is a single point, prove that T is norm-to-norm upper semicontinuous at x if and only if lim diam T[B(x; 0)]
6 _0+
=
O.
2. Monotone operators, sub differentials and Asplund spaces
19
Proposition 2.5. If f i8 a continuOU8 convex function on the open convex 8ub8et D of E, then the 8ubdifferential map x -+ 0f( x) i8 norm-to-weak* upper 8emicontinuou8 on D . Proof. We must show that if x E D and W is any weak* open subset of E* containing of (x), then for any sequence {xn} C D with Xn -+ x, we have of(xn) c W for all sufficiently large n . If not, then there exists a subsequence ( call it {xn}) and x � E of(xn)\ W . By local boundedness of the subdifferential map, we can assume that there is a ( weak* compact ) closed ball which contains the sets of(xn) for all sufficiently large n. Let x· be a weak* cluster point of the sequence {x� }j it is easily verified that x* is in of(x) \ W , a contradiction. Lemma 2.6. Supp08e that f i8 continuou8 and convex on a nonempty open convex 8ub8et D of E and that it i8 Frechet differentiable at a point x in D. Then the 8ubdifferential map of i8 n.orm-to-norm upper 8emicontinuou8 at x. Proof. We want t o show that given any norm open neighborhood V of the functional x* = f' (x) there exists a neighborhood of x which is mapped into V by of. If this were to fail, we could choose 10 > 0, a sequence of points {x n } C D and x� E of(x n ) for each n such that Il xn - x II -+ 0 while II x� - x* 1I > 210. Consequently, there would exist Zn E E, IIz n ll = 1, such that (x� - x·, zn) > 2 10. By Frechet differentiability of f at x there would exist D > 0 such that x + y E D arid f(x + y) - f(x) - ( x*, y) :::; € lIyll whenever lI y ll :::; D. Since x� E of(xn), we have (x�, (x + y) - xn ) :::; f(x + y) - f(xn) so (x�, y) :::; f(x + y) - f(x) + (x�, Xn - x) + f(x) - f(xn) whenever Il y ll :::; D. Let Yn = DZn , so IIYn ll = D and :::; [f(x + Y n ) - f(x) - (x · , Yn )] + (x�, Xn - x) + f(x) - f(xn ) :::; :::; €D + (x�, xn - x ) + f(x) - f(xn). Since of is locally bounded and I (x�, Xn - x ) 1 :::; II x� II ' lI xn - x II , this term converges to 0, while f( x) - f( xn) -+ 0 since f is continuous. But this would yield 2€D :::; €D, an impossibility which completes the proof. Definition 2.7. A 8election
0 satisfying f(x + Dy) + f(x - Dy) - 2f(x ) < 1 /n . D lI y ll=l sup
Since D is a Baire space and since G = nGnU) , we conclude that for some m > 0, the set GmU ) is not dense in D, that is, there exists a nonempty open subset U C D \ GmU ) . We will next define by induction an increasing sequence {Fk } of separable subspaces of E. (The desired subspace F will be the closure of their union.) First, choose Xl E U. It follows (take D = Iii, j = 1 , 2, 3, . . . and use the monotonicity of the difference quotients for J) that there exists a sequence { YI ,j } with II Yl ,j l l = 1 , such that for all D > 0, SUPj
f( Xl + DYI ,j) + f(x! - DYl ,j) - 2f ( x t ) > _ 1/2m. D
Let Fl denote the closed linear span of X l and { Yl ,j } . Clearly, Fl is separable and Fl n U is nonempty (since it contains Xl ) . Given an increasing sequence of separable subspaces Fl , . . . , Fk , define FH l as follows: Let {Xk ,p } , p = 1 , 2, 3, . . . be a countable dense subset of Fk n U and for each p choose a sequence { YP ,j } with II Yp,j ll = 1 such that for all D > ° ( Xk ,p + DYp,j ) + f( Xk ,p - DYp ,j) - 2f(Xk ,p ) > 1 /2m sup). f D _
.
Let Fk+1 be the closed linear span of Fk and {Xk,p } U { YP ,j } , p, j � 1 . Let F = UFk ; then {Xk,p : k,p = 1 , 2, 3, . . . } is a dense subset of F n U . It is clear
24
Section 2
that for each k , p, the point Xk,p is not in G2m UI F) and - since U c D - the sequence {X k ,p } is in D n U, that is, {Xk,p } is a subset of (D n U) \ G2 m U I F). Since G2 mU I F) is open in F, we must have F n U c (D n U) \ G2 mU I F), which implies that f l F fails to be Frechet differentiable at each point of the relatively open nonempty subset F n U of F n D.
Corollary 2.15. A Banach space E is an Asplund space if every separable subspace of E has separable dual. It is immediate from this that every reflexive Banach space is an Asplund space.
2.16 Exercise. Show that for any set r, every separable subspace of co(r) has separable dual, so co(r) is an Asplund space. ( Recall that co(r) is the supremum-normed space of all functions x = (xl' ) on r such that for all € > 0, there exists a finite set r< c r such that I XI' I < € for every I in r\r o'A( x * ) - a } where x * E E*,
a
> 0 and O'A(X* ) = sup{ (x* , x) : x E A}.
Fig. 2.3 If A c E*, then we can also define a weak* slice of A to be a slice where the defining linear functional comes from E (rather than from E** ). For the moment, in fact, we will be working with weak* slices. We say that a nonempty subset A admits slices of arbitrarily small diameter if, for every e > 0 there exists a slice of A of diameter less than e. (This property is equivalent to a concept called "dentability" , so it is sometimes given the same name.)
2. Monotone operators, subdifferentials and Asplund spaces
25
The following lemma might seem a bit peculiar at first, but - as we will see later - its converse is also valid, so it contains a characterization ( a useful one) of Asplund spaces.
Lemma 2.18. If E is an Asplund space, then every nonempty bounded subset A of E* admits weak* slices of arbitrarily small diameter. Proof. Given A, nonempty and bounded in E* , define the sublinear functional p on E by p(x) = O"A(X) == sup{ (x* , x} : x * E A } . Since A is bounded, p is necessarily continuous: p(x) :::; M ll x ll if II x* 11 :::; M for every x * E A. Suppose that there exists f > 0 such that every weak* slice of A has diameter greater than f; we will show that p is nowhere Frechet differentiable. Indeed, given any x E E, for each n � 1 the weak* slice S(x, A, f/ 3n) has diameter greater than f. It follows that there exist x�, y� E S(x, A, f/ 3n) with II x � - Y� II > f, that is x �, y� E A,
(x � , x) > p( x) - f/ 3n,
(y� , x) > p(x) - f / 3n
and (x� - y�, xn) > f for some Xn E E with II xn ll
=
1. Thus
p[x + (l/n)xnl + p [x - ( l/n)xnl - 2p(x) � � ( x � , x + (lin )xn) + (y� , X - ( lin )xn ) =
�
(x � + y� , x) - 2f/ 3n
(l/n)(x � - y� , xn) - 2f/3n > fin - 2f / 3n
=
=
f /3n.
If we divide through by lin it becomes evident ( using Proposition 1.23) that p is not Frechet differentiable at x . We can now characterize separable Asplund spaces.
Theorem 2.19. A separable Banach space E is an Asplund space if and only if E* is separable. Proof. It was shown in Theorem 2.12 that if E* is separable, then E is an Asplund space. Suppose then, that E is a separable Asplund space. If E* is not separable, then its unit ball B* is not separable, hence there is an uncountable set A C B* and n � 1 such that Il x * - Y * II > lin whenever x * , y * are distinct points of A. ( Look at maximal lin-nets in B*.) Since E is separable, B* ( in its weak* topology ) is compact and metrizable, hence satisfies the second axiom of count ability. Thus, A has at most countably many points which are not weak* condensation points; we assume that these points have been deleted from A. Let S be any weak* slice of A. Since such a slice is weak* relatively open and since no point of A is weak* isolated, it must contain two distinct points of A, so its diameter is greater than lin, contradicting Lemma 2.18. We next look at some additional properties which distinguish subdifferen tials within the class of monotone operators, proving some basic results along the way.
26
Section 2
Definition 2.20. A set-valued map T: E -+ 2E* is said to be n-c yclically monotone provided L (Xi , Xk - Xk - l ) � 0 l $ k $n whenever n � 2 and xo , Xl , X2 , . . . xn E E, Xn = Xo , and x i E T(Xk), k = 1 , 2, 3, . . . , n . We say that T is c yclically monotone if it is n-cyclically monotone for every n. Clearly, a 2-cyclically monotone operator is monotone.
2.21 Examples. (a) The linear map in R2 defined by T(X l , X 2 ) = (X 2 , -X l ) is positive, hence monotone, but it is not 3-cyclically monotone: Look at the points ( 1 , 1), (0, 1) and ( 1 , 0). (b) Let / be a continuous convex function on an open convex set; then 8/ is cyclically monotone. (As we will see, this is almost the only example.)
Definition 2.22. A subset G of E x E* is said to be monotone provided (x* - y*, X - y) � 0 whenever (x, x*), (y, yO) E G. If T: E -+ 2E* is a monotone operator then its graph G(T)
=
{(x, x*) E E
X
E*: x* E T(x)}
is a monotone set. A monotone set is said to be maximal monotone if it is maximal in the family of monotone subsets of E x E* , ordered by inclusion. We say that a monotone operator T is maximal monotone provided its graph is a maximal monotone set. There is an obvious one-to-one correspondence between monotone sets and monotone operators. An easy application of Zorn's lemma shows that every monotone operator T can be extended to a maximal monotone operator T , in the sense that G(T) C G(T).
2.23 Exercises. (a) Prove that a monotone operator T: E -+ 2E* is maximal monotone if and only if the following condition holds: Whenever y E E, y. E E · and
(y. - x ·, y - x) ?:
°
for all x E D(T),
x· E T(x)
then it necessarily follows that y. E T( y ). (b) Prove that if T is maximal monotone, then T(x) is convex set, for every x E E. (c) Show that the operator in Example 2.21(a) is maximal monotone.
The next theorem is due to G. Minty. A much more general result has been proved by Rockafellar and will be presented in the next section ( Theorem 3.24) . We start with a simple and useful result.
2. Monotone operators, sub differentials and Asplund spaces
27
Proposition 2.24. If f is convex on D and continuous at x E D, then for all y E E, d+ f(x)(y) = sup { ( x* , y ) : x* E 8/(x)} and this supremum is attained at some point x* E 8/(x) . Proof. A s shown in Prop. 1.8, if x * E 8/(x), then (x * , y ) ::; d+ /(x)(y) for all y. On the other hand, by Corollary 1 . 7, d+ /(x) is a continuous sublinear functional, so for any y f 0 we can use the Hahn-Banach theorem to find x* E E* such that (x* , z) ::; d+f(x)(z) for all z E E ( so x* E 8f(x) by Proposition 1 .8) and (x* , y) = J+ f(x)(y), which completes the proof. Theorem 2.25. If / is continuous and convex on all of E, then its subdiffer ential map 8/ is maximal monotone. Proof. By Exercise 2.23( a), to show that 8f is maximal it suffices to show that whenever y E E and y* E E* are such that y* (j. 8/(y), then there exist x E E and x * E 8/(x) such that (y* - x* , y - x) < O. To simplify the proof we can replace f by the convex continuous function 9 defined by g(x) = f(x + y) - ( y* , x ) . It is easily verified that one has x* E 8g(x) if and only if x* + y* E 8/(x + y). Thus, if y* (j. 8f(y), then 0 (j. 8g(0) and if there exist x* and x with x * E 8g(x) such that (x* , x) < 0, then for z = x + y and z* = x* + y * we have z* E 8/(z) and ( y* - z*, 'y - z) = (x*, x) < O. Assume, then, that y = 0 and y* = OJ we want to produce x E E and x* E 8/(x) such that (x*, x) < O. By Prop. 1 .25, we know that 0 is not a global minimum for f, so there exists a point Xl E E such that f(O) > f(X 1 )' Consider the convex function h(t) = f(tx 1 ), O ::; t ::; 1 . Its right hand derivative at a point to E (0, 1) is clearly equal to d+ /(tOX 1 )(XI ). Suppose this quantity were nonnegative for each such toj by one form of the mean value theorem ( see, for instance, [ Fl, p. 22]) , this would imply that h(O) ::; h(1), a contradiction. Thus, it is necessarily negative for some 0 < to < 1 , so by homogeneity ( and letting x = tox! ), we have d+ f(x)(x) < O. By Prop. 2.24 above, there must exist x* E 8f(x) such that (x* , x) = d+ f(x)(x) < 0, which completes the proof. 2.26 Example. In a Banach space E define f(x) = ( 1/2)lIxIl 2 and define J = 8f; the maximal monotone map J is called the duality mapping for E. Explicitly,
J(x)
= {x· E E· : (x· , x)
=
IIx · II · lIxll and Ilx· 1I = IIxll} .
Proof. I t is readily computed that d+ f(x)(y) = IIxll . d+ llxll(y). If x = 0 , then d+ f(O)(y) = 0 for all y, hence is linear and therefore 8f(0) = {O}. Suppose, then, that x f. O. We know ( by Prop. 1.8) that x · E 8f(x) if and only if x · ::; d+ f(x), that is, if and only if IIxll- I x · ::; d + llxll, which is equivalent to y. == IIxll- I x · E 811xll, that is, if and only if (y. , y - x) ::; IIYIl - lIxll for all y E E. If, in this last inequality, we take y = x + z, II zll ::; 1 and apply the triangle inequality, we conclude that lIy· 1 I ::; 1. If we take y = 0, we conclude that IIxll ::; (y· , x) ::; lIy·II · llxll, so lIy· 1 I = 1 and (y., x) = IIxll, which is equivalent to what we want to prove. The converse
28
Section 2
I I xl l , then for all y in E, we necessarily have is easy: If " y . " = 1 and (y · , x) ( y. , y - x ) S lIyll - lI x ll, so y. E 811 x l l · If the norm of E is Gateaux differentiable, then obviously J( x) contains only one element; by a slight abuse of notation, we denote it also by J( x ) . =
We now know that subdifferential maps of continuous convex functions are maximal cyclically monotone; we will discuss later the fact that maxi mal cyclically monotone mappings are the sub differentials of certain convex functions. These ( and other ) considerations require a fundamental property of monotone operators which was easy to prove for sub differentials of continuous convex functions ( in Prop. 1 . 11); namely, that they are locally bounded at points in the interior of their domain. Definition 2.27.
a) A monotone operator T: E 2 E• is locally bounded at x E D(T) provided there exist M > 0 and [; > 0 such that I IY * II S M whenever y E B(x; 8) n D(T) and y* E T(y). b ) A subset ( not necessarily convex ) A c E which contains the origin is said to be absorbing if E U { AA A > O } . Equivalently, A is absorbing if for each x E E there exists t > 0 such that tx E A. A point x E A is called an absorbing point of A if the translate A - x is absorbing. It is obvious that any interior point of a set is an absorbing point. If Al is the union of the unit sphere and {O}, then Al is absorbing, even though it has empty interior. -t
=
:
Theorem 2.28. Suppose that T: E -t 2 E• is monotone and that x E D(T) . If x E int D(T), or, more generally, if x is an absorbing point of D(T), then T is locally bounded at x . Proof. By choosing any x · E T( x) and replacing T by the monotone operator y -t T(y + x) - x * , we lose no generality in assuming that x = 0 and that o E T(O) . Thus, we want to show that, under these assumptions, T is locally bounded at O. Define, for x E E, f(x) = sup { ( y * , x - y): y E D(T), lI y ll S 1 and y * E T(y)}
and let C = {x E E: f( x) S I}. As the supremum of affine continuous functions, f is convex and lower semi continuous, and hence C is closed and convex. It also contains the origin: First, since 0 E T(O), we must have f 2:: O. Second, whenever y E D(T) and y* E T(y), monotonicity implies that 0 S (y* - 0, y 0) , so f(O) S O. We claim that the closed convex symmet ric set A = C n (-C) is absorbing, hence, by a standard consequence of the Baire category theorem, is a neighborhood of the origin. It suffices to prove that C is absorbing, so suppose x E E. By hypothesis, D(T) is absorbing so there exists t > 0 such that T(tx) =I- 0. Choose any element u* E T(tx). If y E D(T) and y* E T(y), then by monotonicity -
2. Monotone operators, subdifferentials and Asplund spaces
29
(y * , tx - y) ::; (u * , tx - y) .
Consequently, J(t x) ::; sup{ (u * , t x - y ) : y E D(T), Il y ll ::; 1 } < (u * , t x ) + lI u* II < 00.
Choose 0 < ..\ < 1 such that ..\J (t x) < 1 . By convexity J(..\tx) ::; ..\j( t x) + (1 - ..\)J(O) = ..\j ( t x) < 1,
so ..\tx E C . Thus, A is a neighborhood of 0 and hence there exists {j > 0 such that J(x) ::; 1 whenever II x ll ::; 2 {j . Equivalently, if I I x ll ::; 2{j, then (y*, x ) ::; (y* , y) + 1 whenever y E D(T), IIYI I ::; 1 and y* E T(y ). Thus, if Y E B(Oj {j) n D(T) and y* E T(y), then 2{j IlY* II = sup{ (y* , x ) : II x II ::; 2{j} ::; Ily* II . lI y ll + 1 ::; {j IlY * II + 1,
so lI y* 1 I ::; 1/{j · Note that the foregoing result does not require that D(T) be convex. There are trivial examples which show that 0 can be an absorbing point of D(T) but not an interior point (for instance, let T be the restriction of the subdifferential of the norm to the set Al defined above). Even if D(T) is convex and T is maximal monotone, D(T) can have empty interior, as shown by the following example. (In this example, T is an unbounded linear operator, hence it is not locally bounded at any point and therefore D(T) has no absorbing points.) Example. In the Hilbert space (2 let D = {x = (x n ) E e: (2 n xn ) E (2} and define Tx = (2 n x n ) , x E D. Then D(T) D is a proper dense linear subspace of (2 and T is a positive operator, hence is monotone. We use Exercise 2.23 (a) to show that it is maximal monotone. Suppose, then, that y and y ' are in e and that for all x E D( T) =
o ::; (y '
-
Tx, Y - x )
='
(y' , y )
-
(T x, y)
we want to show that Y E D(T) and that T y and let
=
-
(y', x) + (T x , x ) ;
y ' . Fix
n
� 1,
m
(2.2)
� 1 and a E
R
Since x E D(T), we can expand the right side of (2.2) and cancel a number of terms to obtain
O « Y . , y ) - a 2 n Yn _
...,n+m Y Y k k - aYn •
- "" 1
•
+
Yn Yn + 2 n a 2 . •
Letting m -+ 00 yields a2 n (Yn - a) ::; y� (Yn - a) for all n � 1 . Since this is true for arbitrary a E R, it follows that y� = 2 n Yn for each n. Since Y' E e and Y' = (2"Yn ), we conclude that Y E D(T) and Y' = T(y).
It is conceivable that for a maximal monotone T, any absorbing point of D(T) is actually an interior point. A word of caution i� in order at this point. Our knowledge of the structure of the domain of monotone operators is incomplete, although some things are
Section 2
30
known about D(T) when T is maximal monotone. Rockafellar [Ro2] gives two special conditions under which int D(T) =I 0 for a maximal monotone T; in particular, this is the case if the convex hull of D(T) is assumed to have nonempty interior. Under this hypothesis, incidentally, the interior of D(T) is convex, and for any point x E D(T)\int D(T), the set T(x) is unbounded. (Indeed, by the separation theorem there exists Y· ::/= 0 such that (y. , x) 2: (y. , u) for every point u E D(T). Thus, for every A > 0 and any x· E T(x), u· E T(u ) , we have ( x · + A y· ) - u · , x - u) = ( x · - u * , x - u) + A (y · , X - u ) 2: 0 , by monotonicity. By maximality, x· + A y · E T(x) for every A > 0.) 2.29. Exercises. ( a) Prove that if T is maximal monotone, then T(x) is weak* compact and
convex, for all x E int D(T).
( b) Prove that if T is maximal monotone, then it is norm-to-weak* upper semi continuous on int D(T).
The following theorem is due to P. Kenderov [Ke2]. Note that by Lemma 2.18, the "small slices " hypothesis on E* is satisfied if E is an Asplund space. Theorem 2.30. Suppose that every nonempty bounded subset of the Banach space E* admits weak* slices of arbitrarily small diameter; then for every maximal monotone operator T: E -+ 2E• there is a dense Go subset G of W = int D(T) (which we assume nonempty) such that T is single-valued and norm-to-norm upper semicontinuous at each point of G. Proof. For each ;:::: 1 let Gn be the set of all x E W for which there is a neigh borhood V of x in W such that diam T(V) < lin . Clearly, T is single-valued and norm-to-norm upper semicontinuous at each point of the intersection G nGn• Since W is a Baire space, we need only show that each of the sets Gn is open and dense in W. They are open by their definition, so it remains to show that each Gn is dense. Let x E W and let U be any neighborhood of x in W. From Theorem 2.28 we know that T is locally bounded in W, so without loss of generality we can assume that T(U) is a bounded subset of E*. By hypothesis, there exist z E E and a > 0 such that the weak* slice n
=
S == S(z, T(U), a)
=
{x* E T(U): (x* , z ) > O"T(U) ( Z ) - a }
has diameter less than lin. If x * E S, then x * E T(xI ) for some point Xl E U and Xo == Xl + r z is in U for sufficiently small r > o. We claim that T(x o ) C S. Indeed, if y* E T( x o ), then we have 0 ::; (y*
" - x , xo - X l
)
=
r(y* - x" , z ) ,
which implies that y* E S. Since {x * E E·: (x· , z) > O"T(U)(Z) - a } is weak* open and since (by Exercise 2.29(b)) T is norm-to-we�* upper semicontinu ous, there exists /j > 0 such that B (xo ; /j ) C U and T( y ) C S for any point
2. Monotone operators, subdifferentials and Asplund spaces
31
Y E B( Xo j 6)j it follows that T[B(xo j 6)] has diameter less than lin. This says that Xo E Gn n U , which completes the proof. We can now prove the converse to Lemma 2.18, obtaining a characteri zation of Asplund spaces. We first prove a local extension result for convex functions, proved originally by Asplund ( although the proof given below is simpler).
Lemma 2.31. Supp08e that f i8 continuou8 and convex on the open convex 8et D e E and that Xo E D. Then tJ:ere exi8t8 a neig!!:.borhood U of Xo in D and a convex Lipschitzian function f on E 8uch that f = f in U. Proof. Given n 2:: 1, x E E, define fn to be the "inf-convolution" of f and nil 11 : fn(x) = inf{f(y) + n ll x - y ll : y E D } . ·
·
We need to know that fn > - 00 ( at least for all sufficiently large n). To this end, choose x* E 8f(xo). H n 2:: II x* lI , then for any x E E and y E D we have
f(y) - f( xo) 2:: (x* , y - xo ) 2:: - n ll y - xo ll 2:: -n ll y - x II - n ll x - xo ll , so fn(x) 2:: f(xo) - n ll x - xo ll . Thus, we will assume below that n is large enough that fn > - 00 . It follows easily from the definition that for Xl , x 2 E E, o � t � 1 and any f > 0, one has tfn (x I )+(1 - t)fn(x2 ) 2:: fn(txl + ( 1 - t)x2 ) -f, so fn is convex. Also, fn(x) � f(x) for all x E D ( take y = x in the definition ). Moreover, given u , v E E and f > 0 we can choose y E D such that
fn( u ) > f(y) + n llu - Y II - f . Since
fn(v) � f(y) + n ll v - y ll
we have
fn(v) - fn( u ) < f(y) + n ll v - y ll - [f(y) + n llu - YII - f] � n ll v - u ll + f , which shows that fn(u) - fn(v) � n llu - vII for all u , v . Interchanging u , v proves that fn has Lipschitz constant n. Since 8f is locally bounded, there exists a neighborhood U of Xo and an integer n > 0 such that 8f(U) is contained in the ball nB* . Suppose that x E U and choose any functional x* E 8f(x)j then II x * 1I � n and for all y E D
f(x) � f(y) + (x * , x - y) � f(y) + n ll x - y ll , which implies that f(x) � fn(x)j that is, f = fn in U.
Theorem 2.32. A Banach 8pace E i8 an A8plund 8pace if and only if ev ery nonempty bounded 8ub8et of E* admits weak* slices of arbitrarily small diameter.
32
Section 2
Proof. The necessity portion has already been proved in Lemma 2.18. Suppose, then, that f is continuous and convex on the nonempty open convex set D C E; we need only show that l' exists on a dense subset of D. Given Xo E D and a neighborhood U of Xo , we can assume that U is sufficiently small that the conclusion to Lemma 2.31 holds in U; that is, there exists a convex Lipschitz continuous function 1 on E such that 1 f on U. By Theorems 2.25 and 2.30, there is a dense G6 subset G of points in E at each of which a1 is single valued and norm-to-norm upper semicontinuous, hence any selection for a1 is norm-to-norm continuous. By Prop. 2.8, 1 is Frechet differentiable at each such point. But this implies that f has points of differentiability in U. =
As we will see shortly, this characterization has a number of applications. For instance, it is not obvious that a closed subspace M of an Asplund space E is itself an Asplund space; even if we could extend a continuous convex function on an open convex subset of M to one on an open convex subset of E, the (dense) set of points of differentiability of the extension could fail to intersect the (nowhere dense) set M. The Asplund property is inherited by closed subspaces; however, and Theorem 2.32 can be used to prove it. Proposition 2.33. A closed subspace M of an Asplund space E is itself an Asplund space. Proof. We need only show that any bounded nonempty subset A of M* E* /Ml.. admits weak* slices of arbitrarily small diameter. Without loss of generality, we can assume that A is weak* compact and convex. (Any weak* slice of the weak* closed convex hull of A is also a slice of A.) The quotient map Q: E* -t M* is of norm one, onto and weak*-to-weak* continuous. Suppose that f > O. Let B* be the unit ball of E* ; since Q is an open map, the set Q( B*) contains a neighborhood of the origin in M* . Since A is bounded, this implies that there exists -\ > 0 such that Q(-\B* ) = -\Q(B*) � A. Let C = -\B* n Q -l (A); clearly, C is weak* compact, convex and Q(C) = A. By Zorn's lemma there exists a minimal (under inclusion) set C with these properties; let C1 be such a minimal set. By hypothesis, there exists a weak* slice S == S( X, C1 , a) of C1 of diameter less that f; since S is relatively weak* open, the set A l = Q(C1 \ S) is a weak* compact convex set which, by the minimality of C1 , is properly contained in A . If x i , x i E A \ AI , there exist yt , yi E S such that Q( yi) = x i and
Thus, diam(A \ A I ) :::; f. By the separation theorem, there exists a weak* slice of A which misses AI , and hence has diameter at most f. By Theorem 2.32, again, we conclude that M is an Asplund space. Theorem 2.34. A Banach space is an Asplund space if and only if every separable closed subspace of E has a separable dual.
2. Monotone operators, subdifferentials and Asplund spaces
33
Proof. The sufficiency half of this theorem is precisely Corollary 2.15. The necessity follows from the previous proposition and Theorem 2.19. Problem. It is a difficult open question whether a closed subspace of a weak
Asplund space is itself a weak Asplund space.
At this point we should state the obvious: The property of being an As plund (or weak Asplund) space is pres erved under linear isomorphisms; that
is, replacing the norm in a Banach space by an equivalent norm will have no effect on the differentiability of functions on the space nor on a topolog ical property like being a dense Go . This fact has played an important role in the subject, since many results, starting with Asplund's own theorems, were obtained by showing first that particular classes of spaces admit norms which themselves have good differentiability properties, then using that fact to deduce differentiability for arbitrary continuous convex functions. The close connection between differentiable norms and Asplund spaces is illustrated by the following corollary to Theorem 2.32. Corollary 2.35. If a Banach space E is not an Asplund space, then there exists an equivalent norm on E which is nowhere Frechet differentiable.
Proof. If E is not an Asplund space, then by Theorem 2.32 there exists a bounded nonempty subset A of E* and f > 0 such that every weak* slice of A has diameter greater than f. We will use A to construct a bounded symmetric convex set U* with nonempty interior in E* such that every weak* closed slice of U* has diameter greater than f. From the proof of Lemma 2.18, it will follow that the continuous support function p for U* is nowhere Frechet differentiable. Since U* has nonempty interior, p defines an equivalent norm on E. To construct U* , note that both the convex hull C of A U -A and the sum U* = C + B*, where B* is the unit ball of E* , are bounded, symmetric and have the property that all their weak* slices have diameter greater than f. ( This last assertion for U* makes use of the fact that O"C+B* = O" c + O" B " . ) The set U* has nonempty interior, since B* does.
For one of his Gateaux differentiability theorems, Asplund used the hy pothesis that E could be given an equivalent norm which was itself Gateaux differentiable in a certain strong sense. We will approach this result through maximal monotone operators and another theorem of Kenderov. Definition 2.36. ( a) A norm on a Banach space E is said to be strictly convex ( or rotund) provided there are no line segments in the unit sphere; equivalently, provided Il x ll = 1 = lI y ll and x =I- y imply II AX + (1 - A)y ll < A ll x ll + (1 - A) lI y ll
whenever 0 < A < 1.
34
Section 2
(b) A norm on E is said to be smooth provided that for each x E E with II x ll = 1, there exists a unique element x* E E* such that ( x*, x) = 1 and II x* 1I = 1. This is the same, clearly, as saying that the duality map J(x) ( Example 2.26 ) is a single point for every x =f 0, which in turn is the same as saying that the norm is Gateaux differentiable at every nonzero point of E . It is common terminology to refer to a space as being smooth or strictly convex if its norm has that property.
2 .37 Exercises. ( a) P rove that if the norm in E is such that its dual norm in E· is strictly convex [smooth] , then it is itself smooth [strictly convex] . ( b ) Prove that the norm in E is strictly convex if and only if every convex subset of E has at most one point of least norm. (c ) Show that the norm in Hilbert space is both smooth and strictly convex, but that the norms in Co and £1 are neither. ( d ) Prove that the norm in E is strictly convex if and only if II x + yll < II x ll + l I y ll whenever x and y are linearly independent. It is known [ KI ] , [Tr] that there exist smooth spaces whose duals are not strictly convex; we will see in Section 4 that there also exist strictly convex spaces whose duals are not smooth. The proof of the next theorem uses the following classical fact: For any lower semi continuous real-valued function on an open subset of a complete metric space there exists a dense G6 set of points at which the function is continuous. ( See, for instance, [eh, p.ll1] . )
Theorem 2.38 ( [Ked). Suppose that E admits an equivalent norm whose dual norm is strictly convex and suppose that T: E -+ 2E• is a maximal mono tone map such that W = int D(T) is nonempty. Then there exists a dense G6 subset of W at every point of which T is single-valued. Proof. Assuming that the norm in E* is strictly convex, we know that for each x E W , the nonempty weak* compact set T(x) has at most one point of least norm - in fact - exactly one point of least norm, since dual norms are weak* lower semicontinuous. Define the real-valued function j3 by j3 (x)
=
min{ lI x* lI : x* E T(x)} ,
x E W.
We first show that j3 is lower semicontinuous on W ; that is, for any real number A , the set {x E W : j3( x) > A } is open. Indeed, since the norm in E* is weak* lower semicontinuous, the set V = {x* E E*: II x* 1I > A} is weak* open. From Exercise 2.29( b ) we know that maximal monotone operators are norm-to-weak* upper semicontinuous. Thus, if j3( x) > A, then T( x) C V a.nd hence there exists a neighborhood U of x in W such that T(U) C V, which implies that j3( y ) > A for all y E U.
2. Monotone operators, sub differentials and Asplund spaces
35
We next show that if (3 is continuous at x E W, then T(x) consists of a single point. Indeed, suppose that (3(x) = IIx * l I , x* E T(x), and that there exists y* E T(x), with y* t- x*j necessarily, lIy* 11 > I I x* l I . We can choose y E E, lIyl l = 1 , such that (y. , y) > I I x · 1I = (3(x). By continuity of (3 at x, there exists 8 > 0 such that B(xj 8)
c
W and (3(z) < (y* , y) whenever z E B(xj 8).
Let z x + 8y E B(x ; 8) and let z* E T(z) be such that (3(z) monotonicity, 0 :::; (z· - y* , z - x ) 8 (z· - y., y) , =
=
IIz* ll .
By
=
so (z· , y) � (y* , y) > (3( z ) IIz* lI, which is a contradiction. Finally, we complete the proof by applying the general fact about lower semicontinuous functions mentioned above. =
Corollary 2.39. (Asplund) If E admits an equivalent norm which has strictly convex dual norm, then E is a weak Asplund space. Proof. The proof is similar to that of Theorem 2.32. 2.40. Exercises. (a) Suppose that 1 · ·1 is an equivalent norm on E·, (that is, m · lI x·1 I � I x· 1 � M · lI x·1I for all x·, where 0 < m � M). Prove that 1 · · 1 is the dual of an equivalent norm on E if and only if I · ·1 is weak* lower semicontinuous. (b) Suppose that E is separable, that {xn} is a dense sequence in the unit sphere of E and, for x· E E·, define
Prove that I x · 1 = I Ix · 1 I + I I L(x·)1I 2 defines an equivalent strictly convex dual norm on E· . Hence deduce Mazur's theorem from Cor. 2.39.
We next look at a property, weaker than separability, which has implica tions for both Asplund spaces and weak Asplund spaces. Definition 2.41. A Banach space E is said to be weakly compactly generated (WCG ) provided there exists a weakly compact subset K of E whose linear span is dense in E. Since the closed convex hull of a weakly compact subset of a Banach space is weakly compact, one can always assume that K is convex.
For background (and proofs of some of the assertions which follow) see Diestel's lecture notes [Di].
J.
2.42. Examples. (a) If E is separable or reflexive, then it is WCG. In the first case, let {xn } be dense in E and let J( = {n-1 xn/ ll xn ll } u {O}j this is actually compact. In the second case, let K be the unit ball.
36 e..,.
Section 2 (b) The Banach space co(r), for any set r, is WCG : If, for each 'Y E r, we let denote the usual basis vector, then {e..,. : 'Y E r} U {O} is weakly compact.
(c) The Banach space [1 (r) is WCG (if and) only if it is separable, that is, if and only if r is countable (since weakly compact subsets of [l (r) are norm compact). (d) If 11 is finite, then L1 (1l) is WCG. (Since Loo(ll) C L1 (1l) and the inclusion mapping is weak* to weak continuous, the unit ball of Loo(ll) is weakly compact in Ll (Il) ' ) The [I-product of a countable family of WCG spaces is WCG, hence L1 (1l) is WCG if (and only if) 11 is O'-finite. (e) If E is WCG and there exists a bounded linear operator on E having dense range in the Banach space F, then F is WCG.
The next theorem obviously generalizes Theorem 2.12 ( that if E* is sep arable, then E is an Asplund space ) . Theorem 2.43. If E* is WCG, then E is an Asplund space. Proof. w� will apply Cor. 2.15, by showing that for every separable subspace F of E, its dual F* is separable. Since the latter is isometric to the quotient space E* / FJ.. , and since a continuous linear image of a wce space is wce , we can assume that F* is wce . Let K be a weakly compact subset of F* whose linear span is dense in F* ; it clearly suffices to prove that K is norm
separable. But this is immediate from the following general lemma.
Lemma 2.44. If E is a separable Banach space and if K is a weakly compact subset of E*, then K is norm separable. Proof. Since K is weakly compact, it is also weak* compact and (K, w) and (K, w*) are homeomorphic. Any weak* compact subset of the dual of a sep arable space is weak* metrizable, so (K, w*) is separable, and hence (K, w) is separable. Let A be a countable weakly dense subset of K and let L be
the set of all rational linear combinations of elements of A, so L is countable and hence its norm closure L is norm separable. Clearly, L contains all linear combinations of members of A, hence L is the smallest norm closed linear subspace containing A. Since L is weakly closed, it contains the weak closure of A, that is, it contains K. Thus, K is norm separable, since it is a subset of the norm separable space L. An argument ( due to P. D. Morris ) which uses some of the same ideas as above can be used to prove Kuo's theorem that if E* is isomorphic to a subspace of a WCG space, then E is an Asplund space. This may be found in R. Bourgin's lecture notes [Bou]; since H. P. Rosenthal has shown ( see [Di] ) that there exist subspaces of the wce space £ 1 [JL] ' JL finite, which are dual spaces but are not themselves wce, this is a generalization of Theorem 2.43.
2. Monotone operators, subdifferentials and Asplund spaces
37
The converse to Theorem 2.43 is false: By Exercise 2. 16, co ( r) is an As plund space for every set r, while its dual £ l (r) is not WCG if r is uncount able. We conclude this section with another of Asplund's theorems. Theorem 2.45 ( [Asp] ) . If the Banach space E is a subspace of a WeG space, then E is a weak Asplund space. Proof. By a deep renorming theorem due to Amir and Lindenstrauss (see, for instance, [Di] or [D-G-Z2]) any WCG space admits an equivalent norm having strictly convex dual. (This obviously generalizes Exercise 2.40 (b).) Moreover, the new norm inherited by any subspace is easily seen also to have strictly convex dual, so Cor. 2.39 applies.
Remarks. Theorem 2 . 1 1 was suggested by a remark in the Preiss-Zajicek paper [Pr-Z] ; they proved a sharpened form of generic differentiability for continuous convex functions (in a Banach space with separable dual) and pointed out how it could be used to obtain a generic continuity theorem for monotone mappings. (We have replaced their term "a-angle porous" by "a-cone meager" , feeling that the latter is a bit more descriptive.) They also pointed out that D. Gregory's argument could be used to give the proof of Theorem 2 .14 (that Asplund spaces are "separably determined" ) . There i s a great deal of material about monotone mappings and their applica tions in the two volumes by E. Zeidler [Ze] and in the book by D. Pascali and S. Sburlan [pa-SbJ, mostly for reflexive spaces. (The reader is warned that in the early sections the latter authors sometimes assume-without saying so-that their Banach spaces are reflexive.) The fundamental Theorem 2.28 (on local bounded ness of monotone operators) was first proved in slightly different form by Rockafellar [R0 2 ]. The original proof has been considerably simplified by a number of authors; there is one due to P. M. Fitzpatrick (see [pa-SbJ who have attributed to the latter author one of S. P. Fitzpatrick's early papers), and an even shorter proof, using the uniform bounded ness theorem, in [Dr-LJ. The one we present is a specialization to Banach spaces of a general result by J . Borwein and S. P. Fitzpatrick [Bor-F]. Recently, L. Vesely [private communication] has shown that if D(T) is convex, then a maximal mono tone T fails to be locally bounded at each point of bdry D(T). He first notes a Fact: If T is locally bounded at x E D(T) (definition obvious) then, using maximal monotonicity and weak* compactness, x E D(T). Assume that D(T) is convex. If x E bdry D(T) C D(T) and T is locally bounded at x, then by the Fact, x E D(T). Suppose that x were in the boundary of D(T) and that there were a neighborhood U of x with T( U) bounded. By the Bishop-Phelps theorem there would exist a point z E U n D(T) and a nonzero element w · E E· such that (w·, z ) = sup (w·, D(T)). Now, T would also be locally bounded at z, so (the Fact again) ;; E D(T) and we could choose z· E T(z). The same kind of argument used just before Exercises 2.29 would show that T(z) is unbounded, a contradiction. Since x 'f. bdry D(T), it must be in int D(T). By local bounded ness, we can choose an open set U such that x E U C int D(T) and T(U) is bounded. Thus, T is locally bounded at every point of U, which, by the Fact, implies that U C D(T) and therefore x E int D(T), another contradiction.
Section 3
3. Lower semic ontinuous c onvex functions.
Our differentiability results for convex functions made heavy and consistent use of continuity, but in both the theoretical and applied aspects of convex functions, it is sometimes desirable to weaken this hypothesis. Lower semi con tinuity is precisely what is needed. Uncomfortable as it may seem at first, the subject is best treated by introducing the seeming complication of admitting extended real-valued functions, that is, functions with values in R U {oo}. We adopt the conventions that r . 00 = 00 and ( -r ) . 00 = - 00 if r > 0, and ± 00 = ±oo for all r E R. r
(We won't have occasion to worry about 00 - 00 or 0 . 00 . ) Definition 3.1 Let X be a Hausdorff space and let f: X � R U {oo } . The effective domain of f is the set dom(f) = {x E X: f(x) < oo} . We say that f is lower semicontinuous provided {x E X : f ( x) :::; r} is closed in X for every r E R. This is equivalent to saying that the epigraph of f
epi(f) is closed in X
x
=
{(x, r)
E
E x R: r � f(x)}
R. Equivalently, f is lower semicontinuous provided
f(x) :::; lim inf f(xu) whenever x E X and (xu) is a net in X converging to x. We say that f is proper if dom(f) "I- 0. Our original definition of convexity for a real-valued function on a Banach space E applies without change. Note that if f is convex, then so is dom(f). Also, a function f is convex if and only if epi(f) is convex. This last fact is important; it means that
certain properties of lower semicontinuous convex functions can be deduced from properties of these (rather special) closed convex subsets of E x R. One can view this as saying that the study of lower semi continuous convex functions is a special case of the study of closed convex sets.
3. Lower semicontinuous convex functions.
39
3.2. Examples.
(a) Let C be a nonempty convex subset of Ej then the indicator function Ce , defined by Ce( x ) = 0 if x E C, = 00 otherwise, is a proper convex function which is lower semicontinuous if and only if C is closed.
[This example is one reason for introducing extended real-valued functions, since it makes it possible to deduce certain properties of a closed convex set from properties of its lower semicontinuous convex indicator function. Thus, one can use this example to assert that the study of closed convex sets is a special case of the study of lower semicontinuous convex functions. It's all a matter of which viewpoint is more convenient, the geometrical or the analyt ical. It is best to be able to switch readily from one to the other.] (b) Let A be any nonempty subset of E" such that the weak* closed convex hull of A is not all of E" (or, more simply, let A be a weak* closed convex proper subset of E") and let O"A ( X ) = sup { {x * , x) : x" E A}, x E Ej
then 0"A is a proper lower semicontinuous convex function, called the support function of A. (In Sec. 2 we assumed that A was bounded.)
(c) If f is a continuous convex function defined on a nonem pty closed convex set C, extend f to be 00 at the points of E \ Cj the resulting function is a proper lower semicontinuous convex function.
The next proposition uses completeness of E to show where lower semi continuous convex functions are necessarily continuous. Proposition 3.3. Suppose that f is a proper lower semicontinuous convex function on a Banach space E and that D = int dom(f) is nonempty; then f is continuous on D. Proof. We need only show that f is locally bounded in D , since (as observed after Prop. 1 .6) this implies that it is locally Lipschitzian in D. First, note that if f is bounded above (by M, say) in B( X; 15) c D for some 15 > 0, then it is bounded below in B(x; 15). Indeed, if y is in B(x ; 15), then so is 2x - y and 1 1 f(x) � "2 [f( Y ) + f(2x - y)] � "2 [f(y) + M]
so fey) 2': 2f(x) -M for all y E B(x; 15). Thus, to show that f is locally bounded in D , it suffices to show that it is locally bounded above in D. For each 2': 1 , let Dn {x E D : f(x) � } The sets Dn are closed and D UDn ; since D is a Baire space, for some we must have U == int Dn nonempty. We know that f is bounded above by in U; without loss of generality, we can assume that B(O; 15) C U for some 15 > O. If y is in D, with y =f 0, then there exists J-L > 1 such that z = J-Ly E D and hence (letting 0 < ,\ = J-L- 1 < 1 ) , the set n
=
n
.
=
n
n
v
=
'\z + (1 - '\)B(O; 15)
= y + (1 - '\)B(O; 15)
is a neighborhood of y in D; Fig. 3.1 below illustrates the situation.
40
Section 3
Fig. 3.1.
For any point v
=
( 1 - >.)x + >. z E V (where x E B(O; 6)) we have f(v) � ( 1 - >. )n + >.f(z),
so f is bounded above in V and the proof is complete. 3.4. Examples. (a) The function f defined by f(x) = l/x on (0, 00), f(x) = 00 on ( - 00, 0] shows that f can be continuous at a boundary point x of dom(J) where f( x) = 00 . (Recall that the neighborhoods of 00 in (-00, 00] are all the sets (a, oo], a E R.) (b) Suppose that C is nonempty closed and convex; then the lower semicontin uous convex indicator function Dc is continuous at x E C if and only if x E int C. Thus, if int C = 0, then Dc is not continuous at any point of C = dom( Dc ).
Definition 3.5. Recall that if E is a Banach space, then so is E x R, under any norm which restricts to the original norm on the subspace E, for instance, lI (x, r) 1I II x ll + I r l . Recall, also, that (E x R)* can be identified with E* x R, using the pairing =
( x * , r* ) , (x, r))
=
(x * , x) + r* ' r.
Remark. If a proper lower semicontinuous convex function f is continuous at some point X o E dom(f), then dom(f) has nonempty interior and epi(f) has nonempty interior in E x R. (Indeed, f(x) = 00 outside of dom(f), so Xo cannot be a boundary point of the latter. Moreover, there exists an open neighborhood U of Xo in dom(f) in which f(x) < f(xo) + 1 , so the open product set U x {r : r > f( xo ) + I } is contained in epi(f).) Definition 3.7. (a) The definition of the subdiiJerential 8f for a proper lower
semi continuous convex function f is almost the same function: If x E dom(f), define 8f(x) =
=
{x* E E* : ( x* , y - x ) � f( y )
{x* E E* : (x * , y) � f(x + y )
-
-
as
that for a continuous
f(x) for all y E E}
f(x) for all y E E},
while 8f(x) 0 if x E E \ dom(f). It may also be empty at points of dom(f), shown in the first example below. (b) If x E dom(f) we define d+ f(x) as before: =
as
3. Lower semicontinuous convex functions.
d+ f(x)( y ) = lim C 1 [f(x + ty) - f(x)], o t- +
41
y E E,
recognizing that d+ f( x )(y) = 00 if x + ty E E \ dom(f) for all t > O. (It is also possible to have d+ f(x)(y) = -00; consider, for instance, d+ f(O)(l) when f(x) = _x 1 /2 for x � 0, = 00 elsewhere.) The following important relationship is easily seen still to be valid in this more general situation: For any point x E dom(f), x * E 8f(x) if and only if (x * , y) ::; J+ f(x)(y) for all y E E.
It follows from this that for the example given above (f(x) = --IX for x � 0), it must be true that 8f(0) = 0. In the first example below, one sees that it is possible to have 8f(x) = 0 for a dense set of points x E dom(f). 3.8 Examples. {x
(a� Let C be ��e closed (in fact, compact) convex subset of £2 defined by C £ : I x n l ::; 2 , n = 1 , 2, 3, . . . } and define f on C by
=
E
Since each of the functions x -+ _(2 - n + X n )1 / 2 is continuous, convex and bounded in absolute value by 2 ( - n +1 ) / 2 , the series converges uniformly, so f is continuous and convex. We claim that af(x) = 0 for any x E C such that X n > _2- n for infinitely many n. Indeed, let e n denote the n-th unit vector in £2 . If X· E 8f(x) (so that, as noted above, x · ::; d+ f( x)), then for all n such that X n > _2- n , we have
an impossibility which implies that af(x) = 0. Note that if we make the usual extension (setting f(x) = 00 for x E e \ C), then f is lower semicontinuous, but not continuous at any point of C ( = bdry C). (b) Let C be a nonempty closed convex subset of Ej then for any x E C, the subdifferential 8I5c (x ) of the indicator function I5c is the cone with vertex 0 of all x· E E * which "support" C at x , that is, which satisfy {x * , x }
=
sup { { x * , y} : y
E
C}
== 17 c (x · ).
(Indeed, x · E 8I5c(x) if and only if { x ' , y - x } ::; I5c ( y) - l5c (x ) , while x · attains its supremum on C at x if and only if the left hand side of this latter inequality is at most 0, while the right side is always greater or equal to 0.) (c) The following is an example of a function which is continuous and convex on £2 but is not bounded on the unit ball. Define ¢>(t) for t ;::: 0 by ¢>(t) = 0 if o ::; t ::; 1/2 while ¢>(t) = 2t - 1 if t ;::: 1/2. For x = (Xk) E £2 , define 00
f(x)
=
L ¢>( Ek�n X� ). n= l
(As the supremum of a sequence of continuous convex functions, f is lower semicon tinuous and convex. Since the infinite sum actually has only finitely m any nonzero
Section 3
42
terms, f is finite on all of £2 , hence is continuous by Prop. 3.3. It tends to infinity on the sequence of norm-one elements of the form Xl = 7a = Xn = Xn+l while Xk = 0 if k =/: 1 , n, n + 1 . )
3.9. Exercises. (a) A proper lower semicontinuous convex function f has a global minimum on X if and only if 0 E 8f(x).
E at
(b) Suppose C is a nonempty convex subset of E and that f is a proper convex function on E with dom(t) n C =/: 0; then f i e has a minimum at the point X E C if and only if 0 E 8(t + he )(x). Suppose that f is proper convex function on E. Define its dual function (or "Fenchel transform") f' on E' by r (x ' )
= sup { ( x ' , x) - f (x) : x E E},
x · E E' .
(c) Show that f' is a proper lower semicontinuous extended real-valued convex function on E ' . (d) Show that for all x E E and x·
E
E',
( x * , x } � F (x')
+ f(x),
with equality holding if and only if x · E 8f (x) . (e) Define r' on E" in the obvious way and show that its restriction to E coincides with f if and only if the latter is lower semicontinuous.
C E"
(f) Compute r (that is, describe it as a function on E* without explict reference to f) for each of the following functions f: (i) h (x) = II x l i . (ii) h (x) = � II x ll 2 . (iii) fa = he , where C is a nonempty closed convex subset of E. (iv) h = (TA , where A is a nonempty bounded convex subset of E ' . Suppose that f and 9 are proper lower semicontinuous convex functions on E. Define their in/-convolution (or epi-sum) by (t v9 )( x )
=
inf {f( y) + g(x - y) : y E E} ,
x
E E.
(g) Show that fV9 is a proper lower semicontinuous convex function on domain dom(t) + dom(g).
E
with
(h) Given nonempty closed convex sets C) and C2 , compute bel Vhe2 . (i) Show that if C is a nonempty convex set, then
( heV i l . ' I I )(x)
=
dist( x, C).
(j) Using the definition of dual functions (above), show that (t Vg )*
=
r
+ g'.
3. Lower semicontinuous convex functions.
43
(k) Show that if the convex set dom(J) - dom(g) contains a neighborhood of the origin (in particular, if dom(J) n int dom(g) "1= 0) then (J + 9 t = rVg*. (f) Show that fV9 is continuous at
x
E E if either f or 9 is continuous at
x.
Definition 3.10. A point x of a subset X of E is said to be a support point of X provided there exists x* E E* , x* -=1= 0, such that x* attains its supremum on X at x. Any such x* is said to support X at x, or be a supporting functional of X. (This is not to be confused with the support function ax of X, but there is an obvious relationship: x* supports X at x if and only if x ( x* ) = (x*, x) .) The geometric terminology arises from the fact that a closed hyperplane is said to support X if one of its two closed half spaces contains X and if the hyperplane itself actually intersects X. If x* supports X at x, then H = {y E E: (x* , y) = aX ( x*)} contains x and is just such a hyperplane. O'
It is an easy consequence of the Hahn-Banach theorem (or separation theorem) that if a closed convex set C has nonempty interior, then every boundary point of C is a support point of C. It is not obvious that a nonempty closed convex set with empty interior has any support points. Even if it does, there is a question as to how many support functionals it admits. For instance, even though every boundary point of the unit ball B of a Banach space E is a support point of B , there may be functionals on E which do not attain their suprema (that is, their norms) on B. This obviously cannot happen if E is reflexive (since B is then weakly compact) and, in fact, a deep theorem of R. C. James says that if E is nonrefiexive, then there exists an element of E* which does not attain its supremum on B. (See Diestel's Lecture Notes [Di] for a proof.) The fact that both the support points and support functionals of C are necessarily dense (in appropriate spaces) are the Bishop-Phelps theorems. Now, it is important to know that the sub differential of a lower semicontinuous proper convex function f will be nonempty for at least some points of dom(J). By applying the Bishop-Phelps techniques in E x R to the closed convex epigraphs of lower semi continuous convex functions, A. Br�ndsted and R. T. Rockafellar showed that dom(8J) is, in fact, dense in dom(J). There are by now a number of approaches to the foregoing results and to Rockafellar 's theorem that the sub differential map is maximal monotone; our route will pass through key results by I. Ekeland and S. Simons. Definition 3.11. If 0 < A < 1 define K>. = {(x, r) E E x R: A ll x ll � - r } ;
this is easily seen to be a closed convex cone (similar to the cone used in the Preiss-Zajicek theorem) which opens downward; it is the reflection through the origin of the epigraph of the function A ll · , 11 . Since the latter is continuous, K>. has nonempty interior (containing (0, -1), for instance).
44
Section 3
The following lemma is an E x R version of a classical maximality result due to Bishop and Phelps. It simply asserts that, in the partial ordering defined by J{>. , if a closed set A satisfies a certain boundedness condition, then any point of A is dominated by a maximal point of A. Xo. ro)
Fig. 3.2. This illustrates (3. 1 ) and (3 . 2) below Lemma 3.12. Suppose that A is a closed nonempty subset of E x R, that 0 < >.. < 1 and that inf{ r: (x, r) E A } = O. For any point (xo , ro ) in A, there exists (x, r) E A such that (x, r) E A n [J{>. + (xo , ro)] and
(3.1 )
{(x, r)} = A n [J{,\ + (x , r)] .
(3 . 2)
( Figure 3.2 above illustrates properties (3. 1 ) and (3 . 2).) Proof. It is convenient to define the continuous linear functional R on E x R by R(x, r) = r. Choose (xI , rd E Ao == A n [J{>. + (xo, ro)] such that r l < inf R(Ao ) + 1 .
Continuing by induction, we can choose a sequence {(xn, rn)} such that ( X n+ l , rn +l ) E An
==
A n [J{>. + (xn, rn)] and
r n+ l < inf R(An ) + 1/(n + 1) .
We claim that for every
n,
( a) An +l C An
( b ) diam An
and
--t
O.
To see ( a) , note that J{>. + ( X n +l , rn+J ) C J{>. + [J{,\ + (xn , rn)] = J{>. + (xn, rn);
now interesect both sides with A to get the desired inclusion. To prove (b ) it suffi ces to show that if (y , 8) E A n , then l I y - xn ll + 1 8 - rn l � l i n >.. + li n . Note that 8 � infR( An ) � infR(An d > rn - li n ( the second inequality following from ( a)) and >" Il y - xn l l � rn < li n, so Il y - xn ll < li n >.. and 0 � rn - 8 < li n. Since A and J{>. are closed, completeness guarantees that nA n is a single point, which we denote by (x, r). Since (x, r) E Ao , we immediately have (3 . 1 ). To obtain (3 . 2), note that for each n, -
- 8
3. Lower semicontinuous convex functions.
K)... + (x, r)
C
45
K>. + [K>. + ( x n , rn ) ] = K>. + ( x n , rn),
so if ( y, s ) E A n [K>. + (x, r)], then ( y , s ) E A n for every have (y, s ) = (x , r).
n,
hence we must
[The proof given above goes through if E is merely a complete metric space; one simply replaces the partial ordering defined by K), by defining (x, r) ::; (y, s) if and only if >'d(x, y) ::;
s
-
r.
The Banach space version is easier to visualize geometrically, and we have no need for the more general form.]
As an immediate consequence of this result we obtain Ekeland's variational principle, which can be viewed as saying that if f(xo ) is nearly a minimum value for the lower semicontinuous function f, then a small Lipschitz contin uous perturbation of f attains a strict minimum at a point z relatively close to Xo. (That is, there exists a Lipschitz continuous function g, with small Lipschitz constant, such that f + 9 has a strict minimum at z .) This fact has found application in a wide variety of topics in nonlinear analysis; see Ekeland's survey in [Ek] .
Lemma 3.13 (Ekeland) . Assume that f is a proper lower semicontinuous extended real-valued function on the Banach space E which is bounded below. Suppose that f > 0 and that f(xo) ::; inf{f(x): x E E} + f. Then for any A > 0 there exists a point z E dom(f) such thai (i) and
A ll z - xo ll � f(xo ) - fez), (iii)
(ii)
A ll x - z ll + f(x) > f e z ) whenever x #- z.
Remark. Despite the fact that A appears in the denominator, the estimate in (ii) need not be large for small A; one can employ the great trick - to be used to good effect later - of taking A = -If. Proof. We assume without loss of generality that infEf = 0, so we have f(xo ) ::; f. Put the equivalent norm 2AI I . ·11 on E and apply Lemma 3.12 to the closed set A = epi(f) and the cone K1 /2 (which we denote simply by K) to obtain a point ( z, r) in E x R such that (1) ( z , r ) E A n [K + ( xo , f ( xo ))]
and (2) (( z, r) } = A n [K + ( z , r) ].
From (1) we have 0 :::; fez) :::; r < 00 and A ll z - xo ll � f(xo ) - r :::; f(xo ) - fez) :::; f(xo ) :::;
f,
which yields assertions (i) and (ii). Assertion (iii) is obvious if f(x) = 00. To see its validity in general, note first that if f( z) < r , then we must have (z, r) #- ( z , f ( z )) so from (2) it follows that ( z , f( z )) is not in K + ( z, r) , that ,
Section 3
46
is, 0 > r - fez), a contradiction. Thus, r = fez). From (2) , again, if f( x ) < 00 and x =f z, then (x, f(x)) is not in K + (z, r ), that is, ,xlix - zll > r - f(x)
= fez)
-
f(x),
which was to be shown. The Br0ndsted-Rockafellar theorem is an easy consequence of this lemma, but we first require a definition. Definition 3.14. Let f be a proper convex lower semicontinuous function and suppose x E dom(f). For any € > 0 define the €-subdifferential Bd(x) by Bd(x) = {x* : (x * , y) :5 f(x + y)
-
f( x ) + dor all y E E } .
It is clear that if 0 < €1 < €2 , then B€J(x) c B€2 f(x). We show next that this set (necessarily convex and weak* closed) is always nonempty. Proposition 3.i5. If f is a proper lower semicontinuous convex function on E, then Bd(xo ) is nonempty, for every Xo E dom(f) and every € > o. Proof. Figure 3 . 3 below shows immediately what is happening, but we must check some details.
Xo
Fig. 3.3
Since epi(f) is closed and convex and does not contain (xo, f( xo ) - €), there exists a linear functional (y*, r*) E E* X R such that ((y * , r* ), (xo , f(xo ) - f)) < ((y * , r* ) , (x, r )),
x E dom(f),
r � f(x),
that is (y * , xo) + r* [f(xo ) - €J < (y* , x) + r* · r
if x E dom(f), r � f(x).
Taking x = Xo and r = f(xo ) shows that r* > o. We can assume, in fact, that r* = 1; simply replace y* by y* jr* . If we then let x* = -y* and take r = f(x), we obtain the desired inequality. We will also need the following fact, which is of substantial interest in its own right (especially in convex optimization).
3. Lower semicontinuous convex functions.
47
Theorem 3.16. Suppose that f and 9 are convex proper lower semicontinuous functions on the Banach space E and that there is a point in dom( f ) n dom(g) where one of them, say f, is continuous. Then o(f + g) (x )
=
of(x) + o g(x ),
x E dom(f + g).
(The right side is the usual vector sum of sets.) Remark. It is immediate from the definitions that for x E dom(f + g) (which is identical to dom(f) n dom(g)), one must have of(x) + og( x )
c
o ( f + g)( x ).
This inclusion can be proper. To see this, let E = R2 , let 1 denote the indicator function Oc and g = OL , where C is the epigraph of the quadratic function y = x2 and L is the x-axis. Obviously, C and L intersect only at the origin 0 and it is easily verified that 81(0) = R-e, where e is the vector (0, 1), and 8g(0) = Re, while
8U + g)(O) = R2 f= 81(0) + 8g(0).
Proof. Suppose that X o E o(f + g)( x o ) . In order to simplify the argument, we can replace f and 9 by the functions h ex)
=
f(x + x o ) - f(xo ) - (x � , x) and gl ( X )
=
g(x + x o ) - g(xo),
x E Ej
it is readily verified from the definitions that if Xo E o(f + g)(x o ), then o ( h + gt } (O) and if 0 E Ofl (O) + Ogl ( O ) , then X o E of(xo ) + og(x o ). Without loss of generality, then, we assume that Xo = 0, X o = 0, f(O) = 0 and g(O) = O. We want to conclude that 0 is in the sum of(O) + o g( O) , under the hypothesis that 0 E o(f + g)(O). This last means that o E
(f + g)(x) � (f + g)(O)
= 0 for all x E E.
(3.3)
We now apply the separation theorem in E x R to the two closed convex sets Cl = epi(f) and C2 = {(x, r): r � - g(x)}j this is possible because f has a point of continuity in dom(f) n dom(g) and hence - recall Remark 3.6 Cl has nonempty interior. Moreover, it follows from (3.3) that C2 misses the interior of Cl = {(x, r): r > f(x)}. Since (0, 0) is common to both sets, it is contained in any separating hyperplane. Thus, there exists a functional (x*, r*) E E* x R, (0, 0) f:. (x*, r*), such that
-
(x * , x) + r* · r � 0 if r � f(x) and (x * , x) + r* · r � 0 if r � -g(x). Since 1 > f(O) = 0 we see immediately that r* � O. To see that r* f:. 0, (that is, that the separating hyperplane is not "vertical"), we argue by contradiction: If r* = 0, then we must have x * f:. OJ also (x*, x) � 0 for all x E dom(f) and (x* , x) � 0 for all x E dom(g). This says that x* separates these two sets. This is impossiblej by the continuity hypothesis, their intersection contains an
48
Section 3
interior point of dom(f). Without loss of generality, then, we can assume that r * = 1 and hence, for any x E E, (- x * , x - O) � J(x) - J(O) and ( x * , x - O) � g(x) - g(O), that is, 0 = - x* + x* E oJ(O) + og(O), which completes the proof. Theorem 3.17 (Br�ndsted-Rockafellar). Suppose that J is a convex proper lower semicontinuous function on the Banach space E. Then given any point X o E dom(f), € > 0, A > 0 and any x� E oJ (x o ) , there exist x E dom(f) and x* E E* such that
x* E oJ(x),
Il x - xo ll � € / A and I l x* - x � 11 � A.
In particular, the domain of oj is dense in dom(f). Proof. By hypothesis, (x�, x - x o) � J(x) - J(x o )
define
+ € for all x E E, so if we
g(x) = J(x) - ( x�, x ) , x E E, we see that g is proper and lower semicontinuous, with dom(g) = dom(f). Moreover, g(x o ) � infEg + €, so by Lemma 3.13 there exists z E dom(f) such that A ll z - xo ll � € and A ll x - z ll + g(x) 2: g(z) for all x E E. Letting h e x) = A ll x - z ll (x E E), this last inequality implies that 0 E o(g + h) (z ) = og(z) + oh( z) (by Theorem 3.16, since h is continuous). Thus, there exists z * E og(z) = oJ(z) - x� such that - z * E oh ( z ) = {x* E E* : IIx* 11 � A}. Let x* = z* + x� and x = Zj then x* E oJ(x), IIx* - x � 11 � A and Il x - xo ll � € / A, as required. Some important special cases of the Bishop-Phelps density theorems are easy corollaries. Theorem 3.18 (Bishop-Phelps) . Suppose that C is a nonempty closed convex subset of a Banach space E. Then (i) The support points of C are dense in the boundary bdry C of C . (ii) The support functionals of C are dense in the cone of all those functionals which are bounded above on C . Proof· (i). Suppose that Xo E bdry C and that 0 < € < 1. Let J = be be the indicator function of C. Choose X l E E \ C such that IIx o - x l ii < f and apply the separation theorem to obtain x� E E* , IIx � 1I = 1, such that O'e (x � ) < ( X�, X l ) . This implies that for all x E C
(x � , x ) < ( x � , X l ) = ( x � , x l - x o ) + ( x � , x o )
�
f + (x� , x o )
so ( x�, x - x o) � f = J(x) - J(xo ) + f, that is x� E oJ(x o ). By Theorem 3.17 (taking A = yIf) there exist x E C = dom(f) and x* E oJ(x) (which says that x* attains its supremum on C at x) such that
3. Lower semi continuous convex functions.
Il x - xo ll :::; ..fi and
49
II x * - x � 11 :::; ..fi < 1 .
The last inequality implies that x* :f 0, so it is a supporting functional of C at x. (ii). Suppose that x� E E* is such that x� :f 0 and ae(x�) < 00. Given 0 < < II x� 1 I 2 , choose Xo E C such that ( x�, xo) > ae(x* ) - E. If we again let f = fie, then €
(x � , x - xo) < E = f(x) - f(xo) + E for all x E C,
that is, x� E ad(xo) . By Theorem 3.17 (with .A = ..fi again), there exist x E dom(f) = C and x* E af(x) such that II x* - x� 1 I :::; ..fi < Il x � l I . This last inequality implies that x* :f 0, so the proof is complete. Note that assertion (ii) (above) can be reformulated as a variational prin ciple: If the continuous linear functional x� is bounded above on C, then there
exists a continuous linear functional y* of small norm (namely, y* = x * - x�) such that x� + y* attains its maximum on C.
The version of the Bishop-Phelps theorem most frequently applied in the theory of Banach spaces is the following special case of Theorem 3.20(ii) (above); it is what most authors mean when they refer to the "Bishop-Phelps theorem".
Let E be a Banach space. Then the set of all functionals x* in E* which attain their norms on the unit ball, that is, which satisfy
Theorem 3.19.
(x * , x ) = Il x* 11
for some x E E with
Il x ll = 1 ,
is norm dense in E* . Equivalently, the duality map J has dense range. There are much more general versions of the Bishop-Phelps theorems in the original paper [Bi-Ph], but they have not found wide application. For in stance, one can prove that if a functional strictly separates C from a nonempty bounded set X , then it can be approximated by a functional which supports C and strictly separates C from X . An interesting special case of this result (taking X to be a single point) has the following proposition as an immediate consequence; as we will see, it can also be proved using the present methods. Proposition 3.20. A nonempty closed convex subset C of a Banach space E is the intersection of all the closed half-spaces defined by its supporting hyperplanes.
Proof. We must show that if y E E \ C, then there is a support functional of C which separates C from y. Let d dist(y, C) and use the separation theorem to choose x� E E* of norm 1 which separates C from the ball B(y; d); this implies that ae(x�) = (x�, y) - d. =
Section 3
50
slope = d+f(x)(y)
s eparati ng hyperplane Fig.
:>
x + ty
x
x + y
3.4. The functional Xo * separates C and B(y; d)
Let f denote the indicator function Dc and choose E > 0 sufficiently small such that 0(0 + d + E) < d/ 2. Next, choose Xo E e n B(y; d + E), so (x� , Y - xo ) � lI y - xo ll � d + E. For any x E C we have (x � , x - xo) = (x � , x - y) + (x � , y - xo) �
that is, x� E od ( x o ). By Theorem 3.17 (taking ,\ 0) there exist elements x. E dom(f) == C and x; E of(x .) (that is, x ; supports C at x . ) such that II x . - xo ll � 0 and I I x; - x� 1 I � J€. It follows that for all x E C, =
(x ; , x - y) � (x ; , x . - y) = (x; - x � , x . - y) + (x � , x . - y ) �
� II x ; - x� II ' ( 1I x. - xo ll + Il xo - y ll ) + O'c ( x� ) - (x � , y) �
so
O'
c ( x; )
0, apply Prop. 3.20 to obtain a closed hyperplane in E x R which supports C epi (f) (at (y, j(y)), say) and misses the point (x, j(x) - E) , as shown in Fig. 3.5 below. =
3. Lower semicontinuous convex functions.
51
Fig. 3.5
By using the same reasoning as in the proof of Prop. 3.15, this defines an element yOO E EOO which is in af(y) and which satisfies
(y OO , x - y) + f e y) > f(x) 10, -
precisely what is needed. The next two lemmas lead easily to S. Simons' proof of Rockafellar's max imal monotonicity theorem for subdifferentials. Lemma 3.22. Suppose that f is a lower semicontinuous proper convex func tion on E. If a , (3 > 0, Xo E E and f(xo ) < inf E f + a(3, then there exist x E E and xOO E af(x) such that II x - xo ll < (3 and II xOO I l < a .
Proof. Choose 10 > 0 such that f(xo) - infE f < 10 < a(3 and then choose A such that 101(3 < A < a. 1t follows that 0 E ad(xo) so by Theorem 3.17, there exist x E dom(f) and xOO E af(x) such that II xOO Il � A < a and II x - xo l l � lOl A < (3. With f as in the previous lemma, suppose that x E E (not necessarily in dom(f») and that infE f < f(x). Then there exist z E dom(f) and ZOO E af(z) such that fez ) < f(x) and (z OO , x - z) > O.
Lemma 3.23.
Proof. Fix A E R such that infE f < A < f(x) and let A - f(y) , . K = SUP y E E , y # lI y x II _
We first show that 0 < K < 00 . To that end, let F {y E E: fe y) � A } , so F is closed, nonempty and x tt. F. Since dom(f) =I- 0 , one can apply the separation theorem in E x R to find u OO E EOO and r E R such that f � u OO + r. Suppose that y E E and that y =I- x. If y E F, then =
A - f e y) � A - ( u " , y) - r � I A - ( u " , x) - r l + (u " , x - y)
52
Section 3
hence
).. - f(y) < I).. - (u*, x) - r l + * . Ilu lI lI y - x ll - dist(x, F)
If y i F, then II;�W < O . In either case, there is an upper bound for t�Si? , so K < 00. To see that K > 0, pick any y E E such that J(y) < ).. . Since ).. < f(x), we have y -:f x and K � �;{W > O. Suppose, now, that 0 < f < 1, so that (1 - f) K < K and hence, by definition of K, there exists Xo E E such that Xo -:f x and
).. - f(xo) > (1 - f)K. II xo - x II For z E E, let N(z) = K ll z - x lli we have shown that (1 - f)N(xo) + f(xo) < ).. , that is, (N + f)(xo) < ).. + fN(y). We claim that ).. ::; inf E (N + f) Indeed, if z = x, then we have ).. < J(x) = (N + f)(z), while if z -:f x, then ���Si� ::; K, from which it follows that ).. ::; (N + f)(z). Thus, we have shown that there is a point Xo E E, Xo -:f x, such that (N + f)(xo) < infE (N + f) + d{lI xo - x ii · We now apply Lemma 3.22 to N + f, with (:J = Il xo - x ii and a = fK. Thus, there exists z E dom( N + f) == dom(f) and w* E 8( N + f)( z) such that l iz - xo ll < II x - xo ll and II w* 11 < fK. It follows that l iz - xii > O. From the sum formula (Theorem 3.16), 8(N + f)(z) = 8N(z) + 8f(z), so there exist y* E 8N(z) and z* E 8J(z) such that w* = y* + z*. Since y* E 8N(z), we must have (y*, z - x) � N( z) - N(x) K ll z - x ii . Thus (z *, x - z) = ( y* , z - x) + (w *, x - z) � K ll z - x ll - ll w * II ' II x - z ll > > (1 - f)K ll z - x ii > O. Since z* E 8f(z) , we have f(x) � f(z) + (z*, x - z) > f(z), which completes =
the proof.
If f is a proper lower semicontinuous con vex function on a Banach space E, the its subdifferential 8f is a maximal monotone operator.
Theorem 3.24 (Rockafellar).
Proof. Suppose that x E E, that x* E E* and that x* i 8f(x). Thus, 8(f - x*)(x), which implies that inf E (f - x*) < (f - x*)(x). By Lemma 3.23 there exists z E dom(f - x*) == dom(f) and z* E 8(f - x*)(z) such that (z*, z - x) < O. Thus, there exists y* E 8f( z) such that z* = y* - x*, so that (y* - x*, z - x) < O . o i
In Section 2 we noted that the sub differential 8f of a continuous con vex function f is cyclically monotone, but we did not need the continuity
3. Lower semicontinuous convex functions.
53
hypothesis: If f is a convex proper lower semi continuous function on E, if xo , Xl , . . . , X n Xo are in D(8f) and if xi E 8f(Xk), k = 1 , 2, . . . , n, then =
Thus, 8f is both cyclically monotone and maximal monotone. Definition 3.25. A monotone operator T is said to maximal cyclically mono tone provided T = S whenever S is cyclically monotone and G(T) C G(S).
Clearly, a maximal monotone operator which is cyclically monotone is neces sarily maximal cyclically monotone.
It is an interesting fact that subdifferentia� are the only maximal cyclically
monotone operators.
Proposition 3.26 (Rockafellar). 1f T: E -+ 2 E * is maximal cyclically mono tone, with D(T) =I- 0, then there exists a proper convex lower semicontinuous function f on E such that T = 8f.
Proof. Fix Xo E D(T) and Xo E T(xo). For x E E define f(x) = sup { (x;" X - x n ) + (x� _ l ' X n - X n - l ) + . . . + (X � , X l - xo)} where the supremum is taken over all finite sets of elements Xk E D(T) and xi E T(Xk), k = 1, 2, . . . , n, n = 1 , 2, 3, . .. Since f is the pointwise supre .
mum of a family of continuous affine functions, it is convex and lower semi continuous and f(x) > 00 for all x. To see that f is proper, we can use monotonicity of T to show that f( xo) ::; O . Indeed, given any sum of terms of the form entering into the definition of f(xo) , let Yk X n -k and Yk x � k' k = 0, 1, . . . , n. The resulting sum is now the negative of a typical cyclic sum, hence is at most equal to O. By the cyclic maximality of T, to conclude that T = 8f we need only show that G(T) C G(8f) Suppose, then, that X E D(T) and X* E T(x). We will have (x, x*) E G(8f) if we can show that -
=
=
_
.
(x*, Y - x) ::; f(y) - A for all y E E, whenever A < f( x). ( Note that by taking y = Xo , this will imply that - A � (x*, xo - x) when ever A < f(x), which shows that f(x) < 00.) Now, by definition, there exist Xk E D(T) and xk E T(Xk), k 1 , 2, . . . , n, such that =
Let X n+ l = x, x�+ I
x*. For any y E E f ( y) � (X�+ l ' Y - X n+ l ) + (x�, Xn + l - X n ) + . . . + (x � , X l - XO) > (X� l ' y - X n l ) + A (X* , y - x) + A, + + =
=
which completes the proof.
>
54
Section 3
It is obvious that if f and g are proper lower semicontinuous functions which only differ by an additive constant, then af ago Rockafellar [R03] (see also [Tay]) has proved the converse; in particular, this shows that if T is a maximal cyclically monotone operator (so that T = af for an appropriate =
I), then - within an additive constant - the latter is unique.
Consider the following assertion concerning two maximal monotone oper ators S and T:
If D(S) n intD(T) =f 0, then S + T is maximal monotone. (3.4) (By definition, D(S + T) = D(S) n D(T)). Theorem 3.16 (that the subdiffer
ential of the sum of two convex functions is the sum of their sub differentials ) shows that (3.4) is true whenever S = 8f and T = 8g, where f and g are proper lower semicontinuous convex functions. Indeed, if intD(T) =f f.), then [since int dome 8g) C int dom(g) and g is continuous on int dom(g)] we con clude that g is continous at some point of D( S) n D(T) C dom(f) n dom(g), so Theorem 3.16 implies that S + T = 8(f + g) and Theorem 3.24 implies that the latter is maximal monotone. The remark following Theorem 3.16 shows that, even in a two dimensional Banach space, the conclusion of (3.4) can fail if D(S) n intD(T) 0; in that example, the graph of af + ag is strictly contained in that of the maximal monotone operator a( f + g). Rockafellar [Ros] has shown that (3.4) is indeed true for arbitrary maximal monotone operators S and T provided E is assumed to be reflexive, a fact that has been very useful in nonlinear analysis. Its validity for nonreflexive spaces remains an interesting open question. We conclude this section by taking a brief look at a class of maximal monotone operators which do not arise from convex functions (that is, are not cyclic), but which do arise as subdifferentials of a certain class of functions. =
Definition 3.27. (a) Let E and F be Banach spaces. We say that a mapping K : E x F -+ R U {±oo } is a saddle function provided it is concave in the first
variable and convex in the second; more precisely, we require that the function x -+ -K(x, y) [resp. y K(x, y)] be a convex function for each fixed y E E [resp. each fixed x E E]. (b) The domain or effective domain dome K) of K is defined to be the set of points (x, y ) where K is finite-valued and -+
K(x', y) < 00 for all x' E E and K(x , y') > - 00 for all y' E E. We say that K is proper if its domain is nonempty. (c) If K is a proper saddle function we define its subdifferential 8K in the following way: Let a1K(x, y) c E* be the sub differential of the convex function -K(·, y) at the point (x, y) E dom(K) and let fhK(x, y) C F* be the sub differential of K (x, . ) at the same point. For any (x, y) in E x F define 8K(x, y)
=
a1K( x , y) x 82K(x, y) c E* x F*.
3. Lower semicontinuous convex functions.
55
Note that if we put any reasonable norm on E x F, then it is a Banach space ( take, for instance, II (x, y) 1 I = II x ll + Il y lI ) and its dual can be identified with E* x F* , using the pairing ((x * , y* ) , (x, y»)
= (x * , x) + (y * , y) .
3.28 Examples. ( a) Let E = F be Hilbert space and define K(x, y ) = ( 1/2) [ lIyI 1 2 - lI x l n In view of the fact that the derivative of ( 1 /2)11 ' . 11 2 at u is the functional (u, . ) , that is, can be identified with u itself, we see that 8K(x , y) = {(x, y)}.
( b ) Again, let E = F be Hilbert space and define K(x, y) = IIyl 1 2
-
(x , y).
Here we have 8d{(x, y) = { y} and 8d{(x, y) = {2 y - x}, so
8K(x, y) = {( y , 2y - x)} .
Proposition 3.29. If K : E x F --+ R U {±oo } is a proper then its subdifferential oK : E x F --+ 2 (E x F) * is monotone.
saddle function,
Proof. We must show that ((x � , yn - (x; , y; ) , (X I , YI ) - (X 2 , Y2 ») 2::
0
whenever (Xt, YI ) , (X2 , Y2 ) E E x F and that (xi, yi) E OK(Xi, Yi ), i = 1 , 2. Equivalently, given the definition of the pairing between E x F and E* x F* , we want that is,
(3.5)
Since xi E oI K (Xi, Vi ) and yi E 02K (Xi, Vi ), i = 1, 2, we have for all points (x, y) E E x F (3.6) (x � , x - Xl) � -K(x, yt } + K (X I , YI ), (3.7) (X ; , X - X 2 ) � -K(x, Y2 ) + K(X2 , Y2 ),
(3.8) (y; , y - YI ) � K (X I , Y ) - K (x t , y t } , (3.9) (Yi , Y - Y2 ) � K (X2 , y) - K (X 2 , Y2 ) ' We want to evaluate these four inequalities at X2 , X l , Y2 and YI respectively
and then add them to get (3.5) on the left side and 0 on the right side, but we must first check that the resulting cancellations on the right side do not
56
Section
3
involve adding 00 to - 00. By choosing (x, y) to be any element of dom(K), we know-by definition-that - K(x, yJ ) < 00, so (1) implies that K(Xl ' Y l ) > - 00. Also, K(Xl ' Y ) < 00, so (3.8) implies that K(Xl , yt } < 00. Similar reasoning using (3.7) and (3.9) shows that K(X2 , Y2) is also finite. Now, carry out the substitutions indicated at the beginning of this paragraph; it follows that K(X2, Yl ) and K(X l , Y2 ) are also finite valued and we can now sum all four inequalities to get the desired result. Under appropriate topological hypotheses on K (that it be proper and "closed" in a certain reasonable sense) 8K will be maximal monotone; see
[R04] .
Definition 3.30. Let E be a Banach space and suppose that K is a proper saddle function on E x E. Define TK: E --+ 2 E * by
X * E TK(X) provided (x* , x * ) E 8K(x, x ) . The graph G(TK ) of TK is readily seen to be monotone, since it is just the intersection of G( 8K) with the linear subspace D x D* of the product space (E x E) x (E* x E* ), where D [resp. D*] is the diagonal in E x E [resp. E* x E*] .
3.31 Example. Each of the saddle functions J( in Example 3.28 induces the same monotone operator TK , since TK (x) = x for all x in each case.
In recent work, E. Krauss [KrtJ (see also [Kr2] and references therein) has shown that every maximal monotone operator T: E 2 E* is of the form T = TK for some proper closed saddle function K on E x E. The previous example shows that K need not be unique, and Krauss has devoted consider able effort to showing how K can be chosen to have certain special properties. A different representation of general monotone operators on E (as sub differ entials of convex functions on E x E* ) has been investigated by S. Fitzpatrick [Fi2J, who poses a number of related open problems. --+
Remarks. The elementary properties of lower semicontinuous convex functions and their subdifferentials play a fundamental role in convex analysis (an approach to the calculus of variations and optimization which replaces differentiable functions by convex functions). Thus, any contemporary text on that subject will have some overlap with the early portions of this section (and with the first section). See, for instance, Aubin and Ekeland [Au-Ek] and Ekeland and Temam [Ek-T]. There was some reluctance on our part to abandon the original easily visualized geometric approach to the Bishop-Phelps theorems in favor of the more "analytic" approach, using Ekeland's variational principle. (M. Fabian [Fa2 ] has reversed this approach, showing how to deduce Ekeland's principle and the Br�ndsted-Rockafellar theorem (3.17) from a Bishop-Phelps lemma. ) In the first edition of these notes [Ph4] we stayed with the variational approach in part because it was a key step in a slightly
3. Lower semicontinuous convex functions.
57
cumbersome but very useful result by J. Borwein [Bor d which led to the Bri1lndsted Rockafellar theorem and, especially, to a new proof of Rockafellar's theorem (3 .24) on the maximal monotonicity of the subdifferential map (of a lower semi continuous proper convex function ) . The recent, dramatically simpler proof of the latter theo rem by S. Simons lSi) changed all that. While it still seems desirable first to prove Ekeland's variational principle ( since it is the most immediate and fundamental ex ample of a perturbed optimization theorem ) , it is now possible to omit a great deal of material which had been preparatory to Rockafellar's theorem. Most books which present the latter avoid giving a proof (referring the reader to Rockafellar's second of three proofs [R03 )) and at least one book reproduces Rockafellar's incorrect first proof.
Section 4
4. S mo oth variational principles, Asplund spaces, weak
A � plund spaces.
It is clear that Ekeland's variational principle (Lemma 3.13) is an extremely useful form of the "maximality points lemma" (3.12); it was a key step in a sequence of fundamental results. As shown in Ekeland's survey article [Ek] , it has found application in such diverse areas as fixed-point theorems, non linear semigroups, optimization, mathematical programming, control theory and global analysis. Recall the statement: IT f is lower semicontinuous on E, f > 0 and Xo is such that f( x o ) infEf + then for any >. > 0 there exists v E E such that
�
to ,
>' lI x Q
-
v II � f( x o ) - f(v) � f and f(x) + >' lI x v II > f(v) whenever x =f v. -
One drawback to this result is that, even if f be differentiable, the per turbed version f + >' 11 0 v II will not be differentiable at v. This objection was first overcome by J. Borwein and D. Preiss [Bor-P] (see Theorem 4.20 below). A substantially simpler version was later obtained by R. Deville, G. Godefroy and V . Zizler [D-G-Z1,3J, and that is what we present next. In order to han dle Gateaux and Frechet differentiability (and other kinds) simultaneously, we require the following definitions. -
Definition 4.1.
(a) A bomology on E, denoted by (3, will be any family of bounded sets S whose union is all of E, which is closed under reflection through the origin (that is, S E (3 implies - S E (3), under multiplication by positive scalars and is directed upwards (that is, the union of any two members of (3 is contained in some member of (3). There are many possibilities, but the following choices for (3 are of main interest to us: (i) Denote by (3 = G the Gateaux bornology consisting of all finite symmetric sets. (ii) By H we denote the Hadamard bomology, consisting of all compact symmetric sets. (iii) Let W denote the weak Hadamard bomology, consisting of all weakly compact symmetric sets. (iv) Finally, F denotes the Frechet bomology consisting of all bounded sym metric sets. It is clear that G and F are the smallest and largest possible bomologies.
4. Smooth variational principles, Asplund spaces, weak Asplund spaces.
59
(b) A real-valued function f is said to be (3-differentiable at x and x* E E* is called its (3-derivative at x, if for each S in (3,
f(x + ty) - f(x) x * , y -( ) t
-t
0 as t
-t
0+
,
uniformly for y E S. The symmetry and directed properties of a bomology imply that this right-hand limit is in fact a two-sided limit. We denote the (3derivative of f at x by \7 fd( x); it is clear that, in terms of our earlier notation, \7af(x) df(x) and \7 F f(x) f'(x) . =
=
Definition 4.2. A bump function on E is a real-valued function ¢l which is bounded and has bounded nonempty support supp(¢l) {x E E: ¢l(x) 1= O}. We will say that the Banach space E has property ( Hp) provided there exists on E a bump function b which is (3-differentiable and globally Lipschitzian. It is straightforward to verify that a (3-differentiable function f is Lipschitzian if and only if its (3-derivative x \7 pf( x) is bounded on E. =
-t
Proposition 4.3. If E admits an equivalent (3-differentiable norm (at nonzero points), then it necessarily has property (Hp). R be any C 1 function with bounded derivative and nonempty support contained in the interval [! ,. �] and define ¢l( x) = 'IjJ( I x iI ); this is (3-differentiable and vanishes if II x ll < 1/2 or Il x ll > 3/2. Since the
Proof. Let 'IjJ: R
-t
derivative of the norm (at nonzero points) always has norm one, the chain rule shows that ¢l has bounded derivative. The converse to this proposition fails dramatically: R. Haydon [Ha2] has constructed a compact Hausdorff space X with the property that there exists
a Lipschitzian Frechet differentiable bump function on C(X) but the latter does not even admit an equivalent Gateaux differentiable norm.
Definition 4.4. A function f: E ( -00, 00] attains a strong minimum at y E E if f(y) infE f and ll Yn - y ll 0 whenever Y n E E and f(Yn ) -t f(y)· IT f is bounded on E, we define Il f lloo sup { If(x)l: x E E}. -t
-t
=
=
The following general theorem of Deville, Godefroy and Zizler [DGZ3] will yield some important corollaries, which are obtained by making judicious choices of the Banach space F.
Let E be a Banach space and F a Banach space of continuous bounded real-valued functions 9 on E such that (1) IIg lioo ::; II g llF for all 9 E F. (2) For each 9 E F and x E E, the function y gx (Y) = g(x + y) is in F and II gx llF II g II F .
Theorem 4.5.
-t
=
60
Section 4
(3) For each g E F and a E R, the function y g(ay) is in F. (4) There exists a bump function in F. If f: E ( - 00, 00] is proper, lower semicontinuous and bounded below, then the set G of all g E F such that f + g attains a strong minimum on E is a dense Go subset of F. -t
-t
Proof. It is helpful to introduce a notion which is analogous to the notion of a slice used in Section 2. If g: E ( - 00, 00 ] is lower semicontinuous and bounded below, we define, for any a > 0, the closed set -t
S(gj a) = {x E E: g( x) ::; infE g + a}. Tt. is easy to verify that if a > 0 and g l , g2 are both bounded below and satisfy ( 4.1 ) gl ::; g2 + a /3 and g2 ::; g l + a/3, then S(g l j a/3) C S(g2j a). Define Un = {g E F: diam S(f + gj a) < l/n, for some a > O}. We will show that each of the sets Un is dense and open in F and that their intersection is the desired set G. To see that Un is open, suppose that g E Un , with a corresponding a > O. Then for any h E F such that Il g- h llF < a/3, we have Il g- h lloo < a/3 and hence the functions gl = f +g and g2 = f + h satisfy (4.1 ) . Thus S(f+hj a/3) C S(f+gj a) and therefore diam S(f+h; a/3) < l/n, so h E Un . To see that each Un is dense in F, suppose that g E F and e > OJ it suffices to produce h E F such that II h ll F < e and for some a > 0, diam S(f + g + h; a) < lin. By hypothesis, F contains a bump function b. Without loss of generality, Il b llF < e. By hypothesis ( 2 ) , we can assume b(O) t 0 and therefore that b(O) > O. Moreover, by hypothesis (3), we can assume that supp( b) C B(Oj 1/2n ) . Let a = b(0)/2 and choose Xo E E such that
(f + g)(xo ) < irl (f + g) + b(0)/2. Define h on E by h(x) = - b(x - xo); by hypothesis ( 2 ) , h E F and Il h l lF = II b llF < e and h(xo) = - b(O). To show that diam S(f + g + h; a) < l /n , it suffices to show that this set is contained in the ball B(xo; 1/2n ) , that is, if I I x - xull > 1/2n, then x � S(f + g + hj a), the latter being equivalent to (f + g + h)(x) > iW (f + g + h) + a.
Now, supp ( h ) C B(xoj 1/2n ) , so h(x) = 0 if I l x - xo ll > 1/2n hence
(f + g + h)(x) = (f + g)(x) � iW (f + g) > (f + g)(xo) - a = = (f + g + h)(xo ) + b(O) - b(0)/2 � iW (f + g + h) + a, as was to be shown. Suppose that g E n Un ; we want to show g E G, that is, f + g attains a strong minimum on E. First, for all n there exists an > 0
4. Smooth variational principles, Asplund spaces, weak Asplund spaces.
61
such that diamS(f + g; an ) S lin and hence there exists a unique point Xo E n S(f+g; an ) . Suppose that {yt} C E and that (f+g)(Yk) inf E (f+g) . Given n > 0 there exists ko such that (f + g)(Yk) S inf E (f + g) + an for all k 2: ko , therefore Yk E S(f + g; an ) for all k 2: ko and hence II Yk - xo ll S diam S(f + g; an ) S lin if k 2: ko . Thus, Yk Xo and therefore 9 E G. A simple proof by contradiction show that G c n Un , and completes the proof. -t
-t
The first corollary to this theorem is a version of Ekeland 's variational principle (Lemma 3.13). Whereas the latter produces a perturbation (by a small multiple of a translate of the norm) which attains a strict minimum, this corollary produces a similar perturbation which has a strong minimum. It does not, however, yield any control over the location of the minimum point. Before proceeding, we should illustrate the difference between a strong minimum and strict minimum for such perturbations. Consider, for example, any nonreflexive Banach space E; by James' theorem [Di] there exists a continuous linear functional f of norm one on E which does not attain its norm; that is f(x) < 1 whenever x E E, I I xll :::; 1. [Concrete example: f(x) = E2- n xn on E = co.] Thus, if x "I 0, then f( -x/llxll) < 1, that is f(x) + IIxll > 0, which means that f + 1 1 · 1 1 attains a strict minimum at 0. On the other hand, since IIfll = 1 , there exists a sequence {xn} C E, I Ixn ll = 1, such that f(xn) -+ - 1 and hence f(xnH llxnll = f(xnH 1 -+ 0, although Xn f+ 0.
Suppose that f: E ( - 00, 00] is proper, lower semicontinu ous and bounded below. Then for all € > 0 there exists Xo E E such that
Corollary 4.6.
and the perturbed function x at Xo .
-t
-t
f( x) + € II X - Xo II attains a strong minimum
Proof. Let F be the space of all bounded real-valued Lipschitz continuous functions 9 on E with II g l iF = Il g lloo + Il g ll Lip , where I g(x) - g(Y)1 Il g ll Lip = sup{ II x y ll : x, Y E E, x =f Y } . _
It is straightforward to prove that F is a Banach space which satisfies hy potheses (1) through (3) of Theorem 4.5. To verify hypothesis (4), one can apply the construction in Prop. 4.3 to the norm in E to produce a bounded Lipschitzian bump function. Thus, there exists 9 E F such that Il g i l F < € and f + 9 attains a strong minimum at some point Xo E E. Hence, for all x E E,
Ig(x)1 < €, I g(x) - g(xo) 1 S € II X - xo ll and (f + g)(x) 2: (f + g)(xo). It follows that for all x we have f(x) 2: f(xo)+g(xo) -g(x) 2: f(xa ) - € II X - Xo ll ; that is, x -t f(x) + € II X - xo ll attains its minimum at Xo . Also, f(x) 2: f(xo) + g(xo ) - g(x) 2: f(xo) - 2€
62
Section 4
inf E f 2: f(x o ) - 2 € . To see that X o is a strong minimum, suppose that f( Yn ) - g(x o ) + ellYn - x o ll - 0; since
SO
f( Yn ) - f(xo ) + el l Yn - x o l l
2:
(f + g)(Yn ) - (f + g)(x o ),
we must have Yn - X o . Our next application of Theorem 4.5 uses a different choice for the Banach space F.
Df3 denote the linear space of all bounded Lipschitz continuous real-valued functions g: E - R having bounded ,B-derivative 'Vf3g, provided with the norm Definition 4.7. Let
I l lg l l l = I l g li oo + II 'V f3gl l oo ==
sup{ l g(x) l : x E E} + sup{II'Vf3g(x) l I : x E E}.
It is clear that I I I . I I I makes Df3 into a normed linear space, which will be nontrivial whenever hypothesis ( H(3) is satisfied. Completeness of this space is obvious, until someone insists on the details. They are given in the following proposition. Proposition 4.8.
The space (Df3, I I I · l i D is complete.
Proof· For simplicity, if 9 E Df3, we will write g' in place of 'Vf3g. By the mean value theorem [FIl, for any f E Df3 and x, Y E E we have Ig(x) - g( Y )1 � 119'1100 ' 11x - y l I · Suppose, now, that {gn } is a I I I · I I I-Cauchy sequence in Df3. Then both {gn } and {g�} are uniformly Cauchy, so there exists g : E _ R and h: E - E* such that {gn } converges uniformly to 9 and {g� } converges uniformly to h. The boundedness of the sequence {g�} implies that there is a bound on the Lipschitz constants for {gn } , hence 9 is Lipschitzian. It remains to show that 9 is ,B-differentiable and that g' = h, that is, that for each x E E and S E ,B, the difference quotients g(x + t y) - g(x)
t
converge as t - 0+ to (h( x), y) , uniformly for y E S. To this end, fix x E E and S E ,B and let M = sup{II Y II : y E S} . Define, for t E R, t =I 0 and Y E S, gn (x + ty) - gn (x) .J.. 'l' n (t , y ) =
t
while ¢> n (O, y ) ( g�(x), y). Given > 0, the Cauchy property implies that there exists n such that I l g!,. - g� lI oo < provided m 2: n. The mean value theorem cited above shows that =
t:
t:
(4.2)
4. Smooth variational principles, Asplund spaces, weak Asplund spaces.
63
for m 2:: n, t E R and y E S. Since g n is ,a-differentiable, there is a b > 0 such that l ¢in (t, y) - ¢in (O, y) ::; f for all y E S and 0 < t < b. Combined with (4.2), this shows that if m 2:: n, 0 < t < b and Y E S, then
l ¢im (t, y) - ¢in (O, y)1 ::; (1 + M)f .
(4.3)
By hypothesis, if t =f 0, then
¢in (t, y) --. ¢i(t, y) == g(x + ty; - g(x) ,
while (g� (x), y) --. (h(x), y) . Let m 00 in (4.3) to get l ¢i(t, y) - ¢i n (O, y) l ::; (1 + M)f. Let m --. 00 for t = 0 in (4.2) to get I (g(x), y) - ¢i n (O, y) 1 ::; fM. From the last two inequalities --.
I
h(x + ty; - h(x)
_
(g(x), y)1 ::; (1 + 2M)f
for all 0 < t < b and Y E S, which completes the proof. The next corollary to Theorem 4.5 (which uses F D 13) seems startling until one recalls that the intersection of a countable collection of dense G6 subsets of a Banach space is itself a dense G6. =
Corollary 4.9. Suppose that E satisfies property (Hp) and that for each n, the function fn : E (-00, 00] is proper, lower semicontinuous and bounded below. Then for any f > 0 one can choose 9 E Dp such that II g l i oo < f, IIV'pg lloo < f and each of the functions fn + 9 attains a strong minimum on --.
E.
Suppose that the Banach space E satisfies (Hf3) and that f is a proper lower semicontinuous function on E which is bounded below. Then there exists a constant a > 0 (depending only on E) such that for all 0 < f < 1 and for any Yo E E such that f(yo ) < inf f + af 2 , there exist 9 E Df3 and Xo E E such that a) f + 9 has a strong minimum at Xo b) II g l i oo < f and lIV'f3 g lloo < f c) II xo - Yo II ::; f . Theorem 4.10. (DGZ Smooth Variational Principle)
Proof. Let b E Df3 be a bump function with supp(b) C B(Oj 1) and b(O) 2:: 1. By using an appropriate composition, we can assume that 0 ::; b ::; 1 and b(O) = 1. [Indeed, let ¢i: R --. [0, 1] be a Cl function which is monotonically nondecreasing and satisfies ¢i(0) = 0 and ¢i(1) 1. Then ¢i 0 b has bounded derivative, is 0 outside B(Oj 1) and ¢i(b(O)) 1.] Define 1 a= M = max{lIV' f3 b lloo , I } , 4M ' and =
=
64
Section 4
x - Yo x E E. h e x ) = f(x) - 2aE2 b(-- ), 10 Note that h: E -+ ( - 00 , 00) is lower semicontinuous and bounded below. Thus, by applying Theorem 4.5 to F = Dp, there exists k E Dp with Il k lloo < aE2 /2 and II Vpk lloo < 10 / 2 such that h + k attains a strong minimum on E at xo, say. We show first that II xo - Yo II :::; E. Suppose not; then II xo - Yo II > 10, that is, XO �YO ¢ B(O; 1) hence bCO�YO ) = 0 so h(xo) = f(xo) � inf E f. Also, h(yo ) = f(yo ) - 2aE2 < inf f - aE2 .
E
But (h + k)(xo) :::; (h + k)(yo ) so h(xo) :::; h(yo ) + k(yo ) - k(xo ) :::; h(yo) + 2 11 k ll00 < h(yo ) + aE2 ,
hence inf E f < h(yo ) + aE2 , a contradiction. Next, let x - Yo g(x) = - 2aE2 b( --) + k(x), 10
so 9 E Dp and f + 9
=
h + k attains its minimum at Xo . Furthermore,
II g l i oo :::; 2aE2 11 b ll00 + II k lloo :::; 2aE2 + aE2 /2
0 such that for 0 < < S -+
- 00 , 00
( x * , y) < _
f(x + t Y ) - f(x) t
+ 10
( 4.4)
for all y E S. We write x* E opf( x). It follows from this definition that if ,81 C ,82 , then op2 f(x) C opJ(x). Now, we already have a definition of a subdifferential for a lower semicontinuous convex function f, so it is important to note that this new definition extends the old one. In fact, for such functions we have opf(x)
=
oaf(x) = of(x)
4. Smooth variational principles, Asplund spaces, weak Asplund spaces.
65
for all choices of 13. Indeed, by the previous remark, for any 13 we have 8pf( x) C 8af(x). On the other hand, if x*€8af(x), then for all € > 0 and y E E we have (x*, y) ::; d+ f(x)(y) + so x* ::; d+ f(x) and therefore x* E 8f(x). Finally, note that if x* E 8f( x), then the inequality (4.4) holds for all € > 0, y E E and t > 0; that is, for any 13 €,
8f(x) c 8pf(x). We define 13 -superdifferentials by reversing the inequality (4.4) and replacing by - I'.; we denote the set of all these functionals by 8P f(x). Note that €
The next proposition exhibits the relations between the foregoing notions and f3-differentiability. It also makes (in part (c)) the simple but key observa tion connecting the smooth variational principle with subdifferentiability.
(a) If f is lower semicontinuous, f(x) is finite and both 8pf(x) and 8P f(x) are nonempty, then f is f3-differentiable at x, and "pf(x) = df(x). (b) Suppos e that f is concave and continuous in a neighborhood of x. If 8pf(x) is nonempty, then f is f3-differentiable at x and 8pf(x) = {" pf(x)}. (c) If 9 is f3-differentiable at x and f + 9 attains a minimum at x, then f is f3-subdifferentiable at x (that is, 8pf(x) =f 0).
Proposition 4.12.
Proof. (a) Suppose that xt E 8P f(x) and x� E 8pf(x); then for any y in E and for every I'. > 0 we must have (xt , y) - (x�, y) ::; 21'., so xi = x�. Denote the common value by x*; it follows readily from the definitions that x* = "pf(x). (b) Since f is concave and continuous, the convex function - f has a nonempty sub differential and 0 =f 8( - J)( x) = 8p( - J)( x) = -8P f( x) (by the remarks following Definition 4.11). The hypothesis that 8pf( x) be nonempty therefore implies that f is f3-differentiable at x, by part (a). (c) Note first that for all u E E, (-g)(u) - (-g)(x) ::; f(u) - f(x). Let x* = "p ( -g)(x); then given S E 13 and € > 0, there exists 8 > 0 such that < 8 and y E S imply that o 0 and that ,X > O. Assume further that Xo satisfies
Theorem 4.20 (Borwein-Preiss) .
-+
g (xo)
Q means that (x*, y) > Q for each x* E A.
Let T and D be a8 above. Then (i) For each x E D, the real-valued function y I7T(X, y) i8 subadditive and p08itive homogeneou8 and, for any A > 0, I7,xT(X, y) = AI7T(X, y). (ii) For each x E D(T), sup{ l7 (x, y): lI y ll I } sup{ l7 (x, y): Il y ll ::::; I } = sup{ ll x* lI : x * E T(x)}. Proposition 4.29.
-
=
=
(iii) (yT)(x) i8 a 8ingleton if and only if I7(X, - y) = -17(X, y). (iv) If Xo E D, Y E E and (yT)(xo) i8 a 8ingleton (8ay equal to {Q}), then for all f > 0 there exi8ts a neighbohood U of Xo in D 8uch that (T(U), y} > Q - f. (v) Fixing x E D and y =f 0, letting 1 = {t E R: x + ty E D} and defining f(t) = I7(X + ty, y), t E l,
72
Section 4
the function f is monotone nondecreasing on I (and hence is continuous at all but countably many points of I). Moreover, if f is continuous at to E I, then (yT)(x + to Y ) is a singleton. Proof.
(i) and (ii) are immediate from the definitions. (iii) If (yT)(x) is a singleton, then so is (-y T)(x), with ( -y T)(x) = - O'(x , y); that is, O'(x, -y) = - O'(x , y). On the other hand, if ( yT)(x) is not a singleton, there exists xOO E T(x) such that (xOO , y) < O'(x, y) and hence O'(x, -y) 2:: (xOO, - y) 2:: - O'(x, y). (iv) Given f > 0, let W = {yOO E EOO : (yOO , y) > 0: - f l . This is a weak* open set containing T(xo ), so by the norm-to-weak* upper semicontinuity of T at Xo, there exists an open neighborhood U of Xo in D such that T(x) C W for all x E U, which was to be proved. (v) Note that we are not asserting that the open set I is an interval, but this does not affect our argument. Suppose that t l , t 2 E I with t l < t2 ; then for any xi E T(x + tiY)' i = 1, 2, we have (by monotonicity)
hence (xt , y) ::; (xi, y) . This shows that
f(h ) == sup { (xOO , y ) :
xOO E T(x + tlY)}
in particular, f(td ::; f(t2), so single-valued at x + t o Y. Then 0:
Hence if t E at to.
==
::; inf{ ( xOO , y) :
f is monotone.
xOO E T(x + t2Y)}; (4.5)
Suppose, now, that yT is not
inf { (xOO , y) : xOO E T(x + toY)} < f ( to ).
I, t < t o , then (by (4.5)), f(t) ::; 0: < f( t o), so f is not continuous
The relationship between Gateaux smoothness of the norm and �ingle valuedness of T is exhibited by the following lemma.
Lemma 4.30. Suppose that Xo E D and that Yo E E, II Yo II = 1, Yo T is single-valued at Xo, with value 0:. If SUP {O'T (XO , y ): lIyll =
then
I}
is such that
::; 0: ,
T(xo ) c o: ' 81 1 · · 1 1 ( yo).
In particular, if the norm is Gateaux differentiable at Yo, then singleton.
T(xo )
IS
a
Proof. Suppose that xOO E T( xo); then from Prop. 4.29 (ii), we see that 0: 2:: 0 and Il xOO II ::; 0:. Since ( Yo T)(xo) is a singleton, it follows that (x*, Yo} = 0: and so xOO E o: ' 8 1 1 · · 11 ( yo ).
4. Smooth variational principles, Asplund spaces, weak Asplund spaces.
73
Suppose that E admits an equivalent Gateaux smooth norm, that T is a maximal monotone operator on E and that D == int {x E E: T(x) -=I 0} is nonempty. Then there exists a dense G o subset G c D such that T(x) is a singleton for every x E G.
Theorem 4.31.
Proof. Let PI = 11 · 11 denote an equivalent smooth norm on E. Choose sequences of positive numbers 1 /2 > € 1 > €2 > . . . and /31 > /32 > /33 > . such that .
€k
-+
0,
E/3i < 3
and
.
Ey'€k//3k < 00.
To use the Banach-Mazur game we let D be our Hausdorff space and S the set of points in D where T is single-valued. After player A chooses a nonempty open subset U of D, player B's first choice will always be an open subset of U in which T is bounded. Thus, we may always assume that player A's first choice was an open nonempty set U1 in which T is bounded. We may also assume that sup {p; (x*): x* E T[U1 ]} > O.
[Indeed, if this supremum is 0, then B's strategy is obvious: Since T is single valued (equal to 0) on the entire set U1 , she need only choose Vk = Uk for each k = 1 , 2, 3 , . . . so that nVk = U1 c S. ] Now, let Sl be the unit sphere {x E E : PI (x) = I} defined by PI and define
8 1 = sup {a(x, y): (x, y) E U1
X
Sd .
4.29 ( ii ) , we can also write 8 1 sup {pr (x*): x* E T(x) and x E Ud ; by our earlier assumption, 8 1 > O. From part ( v ) of Prop. 4.29, for any y E E and x E U1 , there exist points of the form x + ty E U1 with t > 0 such that a(x, y) � a(x + ty, y) and yT is single-valued at x + ty. Thus 8 1 = sup {a(x, y): (x, y) E U1 X Sl and yT is single-valued at x}. It follows that there exists (X l , Y1 ) E U1 X Sl such that Y1 T i s single-valued at X l and such that (T(xI ), yI) > (1 - €I )8 1 . By Prop. 4.29 ( iv ) , there exists r1 , with 0 < r 1 < 1, such that B(XI, 2rd C U1 and (T[B(x 1 ; 2rI)], Y1 ) > ( 1 - €I )8 1 ' Define VI = B(x 1 , rI ). For all x E E define q1 (X) = dist (x, RxI) == inf { l i x - A xI lI : A E R} and define a new norm P2 on E by P22 - p21 + /312 q21 ' We now have 8 1 , Y1 , X l , P2 and player B's open set VI , so player A may choose any nonempty open subset U2 C VI ' Using a similar strategy to respond to From Prop.
=
_
74
Section 4
player A's choices U2 , U3 , at every step, player B chooses VI , V2 , . . . by constructing sequences of numbers S I , S2 , . . . , S k , . . . , norms PI , P2 , · · · , Pk , · · · (with dual norms pk) , spheres Sl , S2 , . . . , Sk = {x E E : Pk (X) = I } , . . . containing the vectors YI , Y2 , . . . , Yk , . . . respectively, points X l , X 2 , . . . , X k , . . . and positive radii r l , r2 , . . . , r k , . . such that Vk = B(x k , rk), • • •
.
2 2 Pk = P k-l
2 · 2 - 2 + f3k-l qk-l = P I
(where qk(X) = inf { ll x - '\x k l i : ,\ E R} ,
k-l 2 2 + '" L.J f3j qj , j= l
X E E),
S k = sup {17 ( x , y): ( x , y) E Uk X Sd = sup {pk( x * ): X * E T(Uk)}, Yk T is single-valued at X k and
Since B(x k , 2rk )
Vk-l = B(X k-l , rk-t}, we have r k ::; (lj2)rk-l ::; (lj22 )rk_2 ::; . . . ::; (lj2k-1 )rl < Ij2 k-1
as well as Vk
C
C
Uk
Uk for each k. Also, Pk � Pk-l implies that Sk
so, using Uk
C
C
C
Bk-l = {x E E : Pk-I (X) ::; I},
Uk-l and Prop. 4.29(ii),
S k ::; sup {17 ( x , y): ( x , y) E Uk-l x Bk-d = = sup {l7 ( x , y): ( x , y) E Uk-l x Sk-d = S k-l . Note that Pk (Yk- l ) = Pk-I ( Yk- l ) = 1 and X k E Uk C Uk- I , so necessarily (X k , Yk-l ) E Uk x Sk hence S k � 17(X k , Yk- d , while X k E Uk-l implies that 17(X k , Yk-l ) > ( 1 - €k-t }Sk-l . Thus, S k > ( 1 - €k- t }S k-l . Since S I > 0 and € 1 < 1 , this implies that S2 > OJ by induction, S k > 0 for all k. It follows that the decreasing sequence {s k } converges to some number Soo � O. Also, since the diameters of the sets Vk converge to zero, the intersections of their closures consists of a single point, denoted by Xooj necessarily, Xoo E Vk for all k. In order to be able to apply Lemma 4.30 and complete the proof, we will show that (i) the sequence of norms {Pk } converges to an equivalent smooth norm Poo satisfying PI ::; Poo ::; 2PI , (ii) the sequence of vectors { Yk } is convergent, with limit Yoo satisfying Poo( Yoo) = 1 and (iii) YooT is single-valued at x oo, with value Soo. Assuming that we have proved assertions (i), (ii) and (iii), we next prove the inequality in Lemma 4.30, taking, of course, Xoo and Yoo in place of Xo
4 . Smooth variational principles, Asplund spaces, weak Asplund spaces.
75
and Yo , Soo in place of a and Poo for the nonnj that is, we want to show that a(xoo , y ) :::; Soo whenever Poo( Y ) = 1 . For each k,we have Xoo E Uk and Pk ( y ) -l y E Sk and therefore a(xoo, y) = Pk ( y) . a (xoo, Pk ( y ) - l y ) :::; S k . Pk ( Y ) · This last term converges to Soo ' Poo( Y ) = Soo, so the inequality of Lemma 4 . 30 is satisfied and therefore T(xoo) c Soo ' opoo( yoo). Since poo is smooth, T(xoo) is a singletonj that is {xoo } = nVk c S. We have thus described a winning strategy for player B, so by Theorem 4.23, S contains a dense Go subset (the open set D being completely metrizable). It now remains to prove assertions (i), (ii) and (iii). (i) Since qk :::; P I for each k, we have
p� :::; p�
k- l k-l � ; ; (1 = p + L P q :::; + L P; ) . p� :::; 4 . p� , j=l j= l
hence the increasing sequence {pd of equivalent nonns converges unifonnly on bounded sets to a norm Poo satisfying P I :::; Poo :::; 2PI ' To prove that Poo is Gateaux smooth, note first that each of the functions q% is everywhere Gateaux differentiable. In fact, if q%( y ) = 0, then it is easily seen that oq%(y ) = {OJ. If . q�( y ) > 0, then for some Ao E R,
and hence, for any t > 0 and u E E, 0 :::; r l [qk( Y + tu ) + qk( y - tu ) - 2qk( y)] :::; l :::; r [pI ( y - Ao Yk + tu ) + P I ( Y - AOYk - tu ) - 2pI ( Y - AOYk)] .
(4.6)
Since Gateaux differentiability is characterized by the fact that such difference quotients converge to zero (Exercise 1 .24), Gateaux differentiability of P I at points where it is positive implies that the last term in (4.6) converges to zero, hence so does the first tenn. It now follows by standard arguments that the infinite series defining p� is everywhere Gateaux differentiable, hence Poo is Gateaux differentiable at nonzero points. (ii) To show that { Yd converges, we will show that for all k,
since the series EV€k/ Pk converges, this will suffice to show that { Yk } is a Cauchy sequence, hence is convergent to Yoo, say. To obtain the estimate above, recall first that S H I > (1 - fk )S k and f k > f k +I so
a(x HI ' YH . ) > ( 1 - f H . )S H I > (1 - f H . )( l - fk)S k > (1 - 2fk)S k > 0, while - since (X H I , Pk (Yk+I ) - I YH I ) E Uk X Sk - we have
Section 4
76
I 0 we conclude that Ak > 0 for all sufficiently large k . It follows that our earlier estimates on IAk l become 1 - 4Ft/13k � Ak � 1 + 4Vfk/j3k , that is, 1 1 - Ak l � 4Vfk/j3k . Consequently, using the fact that I I Yk l l = PI (Yk ) � Pk(Yk) = 1, we finally obtain the desired estimate:
IIYHI - Yk ll = II(Ak - 1)Yk + uk l l � I Ak - 1 1 · IIYk ll + I lu k l l � � IAk - 1 1 + Iluk ll � 4 ,j€k/j3k + 2 .jfkjj3k = 6 ,j€kjj3k . The fact that Poo (Yoo ) = 1 is immediate from the fact that Pk on bounded sets, since Pk(Yk ) = 1 and Yk --+ Yoo .
--+
P oo uniformly
(iii) Finally, we show that YooT is single-valued at Xoo , with value Soo. To that end, suppose that X* E T( xoo). Since Xoo E Uk for all k and since 1 = Poo (Yoo) � Pk (Yoo), the fact (proved earlier) that
Sk = sup { O"A(X * ) - a}.
If A is a nonempty subset of E*, we define a weak* slice analogously, with the functional coming from E rather than from E** . (b) We say that a nonempty subset A of E is dentable provided it admits slices of arbitrarily small diameter; that is, for every E > 0 there exists x* E E* and a > 0 such that diam S(x* , A, a) < E. (c) If A is a nonempty subset of E*, we say that it is weak*-dentable provided it admits weak*-slices S(x, A, a) of arbitrarily small diameter. Using this terminology, Theorem 2.31 can be reformulated to say that a Banach space E is an Asplund space if and only if every nonempty bounded subset of E* is weak* dentable.
Definition 5.2. A subset A of a Banach space E is said to have the Radon Nikodym property (RNP) if every nonempty bounded subset of A is dent able.
80
Section 5
Since weak* dent able sets are obviously dent able, Theorem 2.31 implies that if E is an Asplund space, then E* has the RNP. One of the most striking results in this area is that the reverse implication is also valid. To that end, we need another definition.
Definition 5.3. An infinite tree in a Banach space E is a sequence {xn} such that Xn = (1j2)(X2n + X2n+ I ) for each n . An infinite tree such that I IX2n - x2n+ I I I 2: 215 for some 15 > 0 and all n is called an infinite h-tree. Ge o metri c P i c t ure
Schemati c Pi cture
"-./ V V "V
X,V XV V )( 2
)( 3
Fig. 5.1 Here is an obvious but important observation: An infinite h-tree cannot be dentable (since every slice must have diameter at least 215). What we will
do is use the characterization that E is an Asplund space ( if and) only if every separable subspace of E has separable dual ( Theorem 2.33) in order to produce a h-tree in E*. We need some preliminary results.
Lemma 5.4. Assume that E is a Banach space which contains a separa ble subspace F such that F* is nonseparable. Then there exists f > 0 and a nonempty subset A of the unit ball B* of E* such that every nonempty relatively weak* open subset of A has diameter greater than f.
Proof. Let B( F* ) be the unit ball of the nonseparable space F* . By the proof of Theorem 2.19, there exist f > 0 and an uncountable subset A l of B(F* ) such that each point of Al is a weak* condensation point of Al and such that I I x* - y* I I > f whenever x*, y* are distinct points of A I . It follows that any nonempty relatively weak*-open subset of the weak* closure A2 of Al contains at least two distinct points of A I . The restriction map R: E* -+ F* maps B* onto B(F* ) and is weak*-to-weak* continuous. Let A be a minimal weak* compact subset of B* such that R(A) = A2 • Thus, if U is a relatively weak* open subset of A, then the image A3 under R of the weak* compact set A \ U is a proper closed subset of A2. Since A2 \ A3 is a relatively weak* open subset of A 2 it contains at least two distinct points of Al and hence there exist distinct points x*, y* in U with II x* - y* I I > f.
5. Asplund spaces, the RNP and perturbed optimization.
81
Let {J{n } be a sequence of nonempty compact convex subsets of a topological vector space such that J{2 n U J{2 n+ l C J{n for all Then there exists an infinite tree {x n } such that X n E J{n for all
Lemma 5.5.
n.
n.
Proof. Let J{ = IIJ{n be the (compact) cartesian product x (x n ) such that X n E J{n for each Define, for each n.
=
consisting of all
n,
Each A n is closed, hence compact, and a sequence {x n } corresponding to an element x E J{ will be an infinite tree if x E nA n , that is, we need only show that this intersection is nonempty. By compactness, it suffices to prove that A l nA2 n · · ·nAk t 0 for each k. To this end, fix k and define x in J{ as follows: IT n > k, let X n be any element of J{n ' From n = k to n = 1 , use induction: Suppose that, for m = n + 1 , n + 2, . . . , a point X m E J{m has been chosen, and define X n = (lj2)(x2 n + X2 n+ l )' Clearly, X n E J{n since the latter is convex and J{2 n U J{2 n + 1 C J{n ' The resulting element x is in A l n A 2 n . . . n Ak, so the proof is complete.
Suppose that E is a Banach space and suppose that there exists a nonempty bounded set A C E* and an 10 > 0 such that diam U > 10 whenever U is a nonempty relatively weak* open subset of A. Then its weak* closed convex hull w* -cl coA contains an infinite Ej2-tree.
Proposition 5.6.
Proof. We construct a sequence {Un } of nonempty relatively weak* open sub sets of A and a sequence {x n } in E such that (a) II x n ll = 1 for each n , U2n U U2 n + l C Un , = 1, 2, 3, . . . , and (c) X* E U2 n and y* E U2 n+ 1 imply that (x* - y*, x n ) � for each First , let U1 = A. Suppose that, for some positive integer m, sets Uk have been defined for 1 :::; k < 2m and points X n have been defined for each 1 :::; < 2 m - 1 so that properties (a), (b) and (c) are valid for all such that 1 :::; < 2 m - I . Suppose k satisfies 2 m - 1 :::; k < 2 m . By hypothesis, we have diam Uk > 10, hence there are functionals Zo and zi in Uk such that II Zo - z i ll > E. Choose points Xk E E such that II xk l l = 1 and (zo - zi , Xk) 10 + 8 for some 8 > O. Let (b)
n
n.
10
n
n
n
=
82
Section 5
Fig. 5.2 Clearly, U2 k and U2 k H are nonempty relatively weak* open subsets of A and U2 k , U2 kH and X k satisfy ( a) , ( b) and ( c ) for n = k. Since n = 2k and n = 2k + 1 exhaust {n: 2m- l < n < 2m}, the construction is complete. We will shortly need the following elementary fact : If It , J2 are nonempty subsets of a topological vector space, then
CoJl - Coh
co(Jl - J2),
C
where co denotes the closed convex hull. [If a E Jl , then the fact that the map x -+ a - x is an affine homeomorphism shows that
a - coJ2 = Co(a - J2)
C
Co(Jl - J2 ).
For any b E coJ2 , we also have COJI b c co( Jl - J2 ), whence the result. ] Now, for each n, let Kn denote the weak* closed convex hull of Un. Each Kn is nonempty, weak* compact and convex, and ( b ) implies that K2n U K2nH C Kn . By Lemma 5.5, there is a tree {x� } in E* such that x� E Kn for each n. By the previous remark, we necessarily have -
X; n - X;n + l E K2n - K2nH
C
w * - cl CO(U2n - U2nH ),
so ( c ) implies that (X2n * - x2 nH , xn ) � f . Hence, I I X2 n - X 2 nH I I � f and {x� } is an infinite f/ 2-tree in Kl = w*-cl coA.
Theorem 5.7.
the RNP.
A Banach space E is an Asplund space if and only if E* has
Proof. As we have observed, Theorem 2.31 contains the "only if" assertion. To prove the converse, suppose that E is not an Asplund space. By Theorem 2.33 there exists a separable subspace F of E such that F* is not separable, hence Lemma 5.4 and Prop. 5.6 together show that E* contains a bounded 8-tree, for some 8 > O. Since the latter is not dent able, the space E* does not have the RNP. If C is a bounded closed convex subset of a Banach space with the RNP (or more generally, if C itself has the RNP ) one can deduce a number of
5. Asplund spaces, the RNP and perturbed optimization.
83
strong implications concerning the extremal structure of C; for instance, such a set is the closed convex hull of its extreme points, a property usually only associated (via the Krein-Milman theorem) with compact-or weakly compact convex sets. More generally, we will eventually show that any such set is the closed convex hull of its strongly exposed points.
Definition 5.8. A point x in a closed convex set C is said to be strongly exposed provided there exists x* ¥- 0 such that x E S( x*, C, a) for every a > 0 and these slices have diameters converging to 0 as a tends to o. Equivalently, x is strongly exposed by x* if, for {x n } C C,
(x*, x n )
--+
ac(x*) implies II x n - xii
--+
o.
A functional x* which satisfies the definition above is called a strongly exposing functional, and is said to strongly expose x. A point x E C i s called exposed if there exists x* ¥- 0 such that
ac(x*) for each y E C, y ¥- x, and x* is said to expose x. If C C E*, a point x* E C is said to be weak* strongly exposed (or weak* exposed) if analogous properties hold, with the exposing functionals coming from E. (x * , x)
= ac(x*) and
(x*, y)
0 and y is fixed, divide by r and let r -+ 00 to get (x*, y) ::; p(y) for all y. Thus x* is in C. Moreover, setting y in (5. 1 ) equal to 0 yields (x* , x) 2: p(x), so equality holds at x. Conversely, if x* ::; p and (x* , x) = p(x), then (5.1) is immediate. It is also easy to see that
5. Asplund spaces, the RNP and perturbed optimization.
85
p(x) = O'c(x) sup {(x* , x ) : x* E C}; indeed, if x E E, then (x*, x ) :::; p(x) for all x* E C, so O'c(x) ::; p(x). Since p is continuous, there exists an element x* E op(x), so x* E C and p(x) = (x*, x ) :::; O'c(x). ==
The duality property in which we are interested is described by the fol lowing proposition.
Suppose that p is a Minkowski gauge on E. An element x* in C(p) is weak* strongly exposed by x E E if and only if p is Frechet differentiable at x, with p' (x) = x* .
Proposition 5.11.
Proof. Suppose that x* E C is weak* strongly exposed by x =f 0, so that (x* , x ) = O'c(x) p(x). Since p is continuous, op(y) i= 0 for all y in E. Let 'P(y) be any element in op(y) if y =f x, and let 'P(x) = x* . By Prop. 2.8, it suffices to show that 'P is norm-to-norm continuous at x. Suppose, then, that X n ---+ x. Since, for all 'P(x n ) E op(x n )' we have 'P(x n ) e C and ('P(x n ), X n ) = p(x n ). Moreover, 1('P(x n ) - x* , x ) 1 = 1('P(x n ) - x* , x - x n ) + ('P(x n ), x n ) - (x *, x n ) 1 ::; ::; 11'P(x n ) - x* II ' lI x - xn ll + Ip(x n ) - (x*, x n) l· Since p(x n ) ---+ p(x) and (x*, x n ) ---+ (x* , x ) p(x), and 11'P(x n ) - x* 1 1 IS =
n,
,
=
bounded, the right side converges to O. Therefore
(x* , x ) = p(x). Since x* is weak* strongly exposed by x, we have 11'P(x n ) - x*11 norm-to-norm continuous at x; moreover, x* = p'(x). ('P(x n ), x )
---+
---+
0,
so 'P is
To prove the converse, suppose that p is Frechet differentiable at x =f O. Let x* = p'(x); since x* E op(x) we have x* ::; p and ( x*, x ) = p( x). Suppose that there exists {x�} C C and r > 0 such that
(x�, x ) ---+ (x *, x) = p(x) but Il x� - x* 1 I There exist {Yn } C E such that ll Y n I = 1 and (x� - x*, Yn ) > r for all
>
r for all
n.
(5.2)
n.
By Frechet differentiability, given 0 < e < r / 2 there exists
0
>
0
such that
+ y) - p(x) - (x* , y ) ::; e l ly ll whenever Il y l l :::; o . Since x� E C we have (x� , x + y) ::; p(x + y) for all y E Let Zn = oYn ; then from (5.2) we have Or < (x � - x*, zn ) = (x � , x + zn ) - (x* , x ) - (x*, zn ) - (x� - x*, x ) ::; :::; [P(x + zn ) - p(x) - (x*, zn ) ] - (x� - x*, x ) ::; or / 2 - (x� - x*, x ) , 0 ::; p(x
E.
86
Section 5
which leads to a contradiction, since (x� - x*, x)
-+
o.
This proposition is the basis for the following dual characterization of Asplund spaces.
A Banach space E is an Asplund space if and only if every nonempty weak* compact convex subset C of E* is the weak* closed convex hull of its weak* strongly exposed points.
Theorem 5.12.
Proof. One direction is immediate from Theorem 2.32: If every such subset C C E* has weak* strongly exposed points, then every bounded nonempty subset A of E* is weak* dent able (since small weak* slices of the weak* closed convex hull C of A yield small weak* slices of A). To prove the converse, suppose that E is an Asplund space and that C is a nonempty weak* compact convex subset of E*. Without loss of generality, 0 E C. Let p = O"c; then p
is a Minkowski gauge and - as the definition and an easy application of the separation theorem show - C(p) C. Let D be the weak* closed convex hull of the weak* strongly exposed points of C and suppose that D � C. Then there exists x E E such that D( x) < c( x). Since C is bounded, both of these convex support functionals are continuous and so there exists a point of Frechet differentiability Xo of p contained in the open set where the strict inequality holds: D( xo) < O'c( xo). By Proposition 5.11, Xo strongly exposes C at the point x· p' C xo) and so (x·, xo) = p ( xo) O'c(xo). But D contains all the weak* strongly exposed points of C, therefore (x·, xo) � O' D(X O ) < O'c(x o ), a contradiction. =
0'
0'
0'
=
=
As another application of Proposition 5.11, we can now give the details of Example 1.14(c). Example l.14(c). There exists an equivalent norm on £ 1 which is nowhere Frechet differentiable but which is Gateaux differentiable at each nonzero point.
Proof By Exercise 2.37(a), to obtain an equivalent smooth norm on £ 1 it suffices to define an equivalent strictly convex dual norm on £00. To that end, define II . · 1 1 on £00 by Clearly, lIyl l oo � lIyll � 2 11Yll for each y E £00 , so this is an equivalent norm. Since 00 p is the supremum of a sequence of weak* continuous functions, it is weak* lower semi continuous, so II . · 11 is a dual norm. As the sum of two norms, one of which (namely, p) is strictly convex, I I 1 1 is easily verified to be strictly convex. It remains · ·
to show_that the norm 1 1 · · 11 which it induces on £1 is nowhere Frechet differentiable, and this will follow from Prop. 5.12 if we show that the unit ball it defines in £00 has no weak* strongly exposed points. Suppose, then, that x E £1 with II x ll = 1 and that y E £00, lIyll = 1, is the unique element such that ( y , x ) = 1. It suffices to produce a sequence {l} in £OCJ such that (y k , x )
-+
( y , x)
=
k 1 and lIy l l
-+
lIyll
=1
5. Asplund spaces, the RNP and perturbed optimization.
87
but lIy k - yll does not converge to zero. There are two cases to consider, depending on whether Yn --+ O. If the latter is the case, then there exists N such that IYn l < 1/4 whenever n > N. For k > N, define y k E f oo by
Yn if n :/:- k, ( y k),.
=
3/8.
(y, x ) - X k · ( Yk - 3/8)
--+
(y , x)
max { 3/8, sup IYn l } n oF "
=
IIYll oo .
( y")n Since X
k
--+
0, we have
(y" , x) Also, 1
=
=
=
1.
=
lIyll $ 2 11yll 00 so lIylloo � 1/2 . Thu s, lIy" lIoo
It follows that for k
=
--+ 00 ,
Ily k ll = IIYlloo + [ET n y� - ( y� - 9/64) . T"j l/2
--+
Ilyl i .
On the other hand, Ily" - y ll � Ily" - ylloo = 1 3/8 - y,, 1 > 1/8. Suppose, now, that {Yn} does not converge to zero. There necessarily exist £ > 0 and a subsequence {Ynk } such that IYnk I > 2£ for all k. Define y k in f oo by
(y")n Then (yk , x)
--+
(y , x)
=
Yn if n :/:- n k , (y ")nk = £. 1 and lIy"lI --+ lIyll as before, while =
lIy - y" 11 � Ily - y" lloo
=
IYnk - £ 1 > £ ,
which completes the proof.
Some of the results in this and earlier sections can be interpreted theo rems in infinite dimensional optimization, in that they give existence results for minimum (or maximum) points for certain types of functions. Actually, what they show is better described as "perturbed" optimization. In finite dimen sional optimization, existence is frequently trivial: When one has a continuous (or lower-bounded lower semicontinuous) real-valued function f on a bounded and closed set C, then, since the latter is compact, the existence of a mini mum point is guaranteed. In the infinite dimensional case, minimum points need not exist, and one will settle for a theorem of the form: For some Lipschitz function h, having Lipschitz norm as small as one may wish, the perturbed function f + h attains its minimum on C. Ekeland's Lemma 3.13 can be cast in this form (as noted just preceding its statement). So can the Br¢ndsted Rockafellar Theorem 3. 17, which has the following consequence : Suppose that f is a convex proper lower semicontinuous function which nearly attains its minimum on the Banach space E at the point x. Then there is a continuous linear functional x * of small norm such that f + x * does attain its minimum. (Moreover, it does so at a point which is close to x . ) In these results, com pactness was replaced by completeness and convexity. Results of this character have been obtained by C. Stegall [Ste2J (who replaces compactness by convex ity and the Radon-Nikodym property) and by N. Ghoussoub and B. Maurey [Gh-M2J (who look at a certain class of subsets of dual Banach spaces and as
88
Section 5
replace compactness by weak* metrizability and weak* relative compactness). We will formulate and prove Stegall's theorem in terms of maximum points for upper-bounded and upper semicontinuous functions. (The theorems of Ghous soub and Maurey will be stated without proof at the end of this section.) Our proof of Stegall's theorem requires a straightforward extension of the notion of "slice" , so as to include certain nonlinear functions. Definition 5.13. Suppose that C is a nonempty set and that f is an upper bounded real-valued function on C. For each > 0 define a
S( f, C, a) =
{x E C : f(x) > sup f - } . e
a
A point x E C is said to be a strong maximum for f if f( x ) = SUPe f and Il x - x nll 0 whenever f(x n ) f(x). (Note that we could have said that f "strongly exposes" x; some authors use this terminology.) �
�
5.14. Exercise. If C is closed and nonempty and f is a real-valued upper-bounded upper semi continuous function on C, then f attains a strong maximum on C if and only if diam S(f, C, o:) ..... 0 as 0: ..... 0 + .
Suppose that C c E is a nonempty closed and bounded convex set with the RNP and that f is an upper-bounded upper semi continuous real-valued function on C. Given € > 0, there exists x* in E* such that Il x* 1 I :::; € and f + x* attains a strong maximum on C. Theorem 5.15. (Stegall).
Remark. An obvious difference between this theorem and Ekeland's vana
tional principle is that while both have the same hypotheses on the function to be minimized (or maximized), Stegall 's result obtains a maximum point after adding a small linear function (which is also Lipschitz, of course). This strengthening of Ekeland's variational principle comes at the cost of assuming that C has the RNP. Before we start the proof of Theorem 5.15 we need to recall the original definition of dent ability, in the form of the following exercise (which is a simple application of the separation theorem). We will also require several lemmas. 5.16. Exercise. Prove that a nonempty closed set A in a Banach space E is dentable if and only if for every € > 0 there exists x E A such that x is not in co( A \ B( Xj €».
The next lemma gives a recipe for constructing nondentable sets; we will use it when we prove Theorem 5.15 by contradiction. Lemma 5.17. Suppose that {A n } is a sequence of (eventually) nonempty subsets of a Banach space E with the following property: There exist constants € > 0 and ), > 0 such that for all x E co A n and y E E,
5. Asplund spaces, the RNP and perturbed optimization.
89
dist [x, co( An+l \ B( y ; E))) ::; >'/2n . Then the set
A = n n U {co Aj: j � n}
is nonempty and not dentable. Proof. We will show that x E co(A \ B(x; E/2)) for each x E A. First however, we show that A is nonempty and that for each n � 1 , (5.3)
where B == B(O; 1 ) . To this end, fix n sufficiently large that An is nonempty and suppose that Xo E co An . By hypothesis, there exists a point X l E co An+ l such that II xo - xI I I ::; 2>./2 n . Similarly, there exists a point X2 E co AnH such that II xI - x2 11 ::; 2>. /2n+ l . Continuing by induction, we obtain Xk E co An+k, k = 0 , 1 , 2 , . . . such that E ll xk - XkH I I < This implies that the series E(Xk - X k + l ) converges to an element y E E, of norm at most 4>'/2n . By writing y as a limit of the (collapsing) partial sums we get y Xo - Z where Z lim xk. It follows that Z E A (so A is nonempty) and Xo E A + (4)./2n )B, which proves (5.3). Suppose that x E A. Fix m such that 4>'/2m < E/2. Since we have x E {Uco Aj:j � n} for all n, for each n � m there exist j � n and Yn E co Aj such that I l x - Yn II ::; >'/2n. By hypothesis, 00.
=
=
dist [Yn , CO(AjH \ B(x; E)) ) ::; >./2j ::; >./2 n , so there exists Zn E co(Aj+1 \ B(x; E)) such that l l Yn - zn l l < 2>./2n. We can write Zn as a finite convex combination Zn
=
E>'j u j,
U j E AjH \ B(x; E).
By (5.3), for each i we have U j E A + (4)./2j )B c A + (4)./2n)B, so there exists Vj E A such that IIU j - Vj II ::; 4>./2n. Let Wn = E>'j v j; it follows that I l zn - w n ll ::; 4>./2n and hence I l x - w n ll ::; 7>./2n . Since Ilu i - x i i � E for each i, we have II Vj - x II
� E - 4>'/2 n � E - 4>./2m > E/2,
that is, Wn E coCA \ B(x; E/2)). It follows that x E coCA \ B(x; E/2)) required. The next two lemmas will allow us to reduce the proof of Theorem 5. 13 to simply showing that there are arbitrarily small perturbations f + X* of f which define small slices. The first lemma shows that we can get decreasing families of slices provided we perturb the function f slightly; the proof is a straightforward verification from the definition. Since all our slices will be in the same set C, we will write S(J, 0) in place of the usual S(J, C, 0). as
Suppose that ihe real-valued function f is bounded above on the nonempiy subset C of the unit ball B of E. For any 0 > 0 we have S(J + x*, (3) c S(J, 0) provided Il x* II < 0/2 and 0 < f3 < 0 - 2 11 x* II ·
Lemma 5.1B.
90
Section 5
Let C be a nonempty bounded closed subset of E and suppose that for every upper-bounded upper semicontinuous function f on C and every E > 0, there exist x* E E*, II x*11 < E, and 0 > 0 such that diam S(f + x*, 0) :::; 2E. Then given E > 0 there exists x* E E* such that II x*1 I < E and f + x* attains a strong maximum on C. Lemma 5.19.
Proof. We assume without loss of generality that C is contained in the unit ball of E and that 0 < E < 1. By hypothesis, there exist xt E E* , II xt I < E/2, and 0 < 01 < 1 such that diam S1 < E, where we write S1 = S(f + xr, 0 1 ) . By applying the hypotheses to f + xr and E1 01 E/22 , we obtain II xi l i < E1 so that diam S2 < 2E1 , where S2 S(f + xi + X i , 02 ) and 0 < 02 < 0 1 · Continuing by induction with 00 1 , we obtain sequences =
•
=
=
En > 0,
0 < O n < 1, x� E E* ,
Sn = S(f + Ef xi,
on)
such that
II x� II < En - 1 , En E on/2 n+ 1 and O n < 0n - 1 · Obviously, the series Ex � converges to a point x* E E* of norm at most E · Eon2-( n +1 ) < E. Note that diam Sn ---+ 0, since En ---+ O. We claim f + x* diam Sn :::; 2En - 1 ,
=
•
attains a strong maximum on C. By Exercise 5.14 it suffices to show that diam S(f + x* , 0) ---+ 0 as 0 ---+ 0+ , and for this it suffices to prove that for all there exists 0 > 0 such that S(f + x* , o) C Sn . This follows from Lemma 5.18, using the fact that f + x* = f + Eix i + w : where n
IIw� I
k :::; E�n+ 1 E Ok - 1 /2 < on/2; •
we need only choose 0 such that 0 < 0 < O n - 2 I 1w :lI .
Proof of Theorem
5. 1 5. By utilizing the previous lemma, we need only show that given any E > 0 there exist x* E E*, II x* II < E, and 0 > 0 such that diam S(f + x*, 0) :::; 2E . Proceeding by contradiction, suppose that for every II x* II < E and each 0 > 0, we have diam S(f + x* , o) > 2E . For each let n
For all sufficiently large we have E - 2 - n > 0, so that An is nonempty. Let >. = 5/2; we will show that with this choice of >., the sequence {An} satisfies the hypotheses of Lemma 5.17, the conclusion of which will contradict the hypothesis of Theorem 5.15 (that C has the RNP). Restating the main hypothesis of Lemma 5.17, we want to show that for any y E E, n,
co An c co(An+1 \ B(y; E)) + (>./2n ) . B.
(5.4)
Since the set on the right side is cunvex, it suffices to prove that it contains An. Suppose, then, that x E An but, for some y E E, it is not in the right hand side of (5.4) (which has nonempty interior). By the separation theorem there exists y* E E*, I y* I = 1, such that
5. Asplund spaces, the RNP and perturbed optimization.
91
(y* , x) � sup { (y*, u ): u E An+! \ B(y; E )} + >'/2 n . (5.5) Since x E An , there exists x* E E* with Il x* 11 ::;: E - 2- n such that x is in the slice S(f + x*, 1/4n ). Write z * = x* + (1/2 n +1 ) y*; it follows that IIz * II ::;: E - 2 - n + 2-( n+ l ) = E - 2-( n+ l ) so S(f + z*, 1/4n+! ) C An +! . Since diam S(f +z*, 1/4n + 1 ) > 2 E , this slice is not contained in B(y; E ) . This implies that there exists z E C \ B(y; E) such that f ez ) + (z * , z) > sup ( f + z *) - 1/4n+ l . (5.6) •
c
Thus, z E A n+ l \ B(y; E) and hence, from (5.5), we conclude that
(5.7) (y * , x) � (y*, z) + >'/2 n . We will prove the inclusion in (5.4) by showing that this cannot be true. Note first that since x E S(f + X* , 1/4n ) and z E C, we necessarily have f(x) + (x *, x) > sup( f + x*) - 1/4 n � fez ) + (x * , z) - 1/4n . (5.8) c
Similarly, from (5.6) we have
fez ) + (z * , z) == f ez ) + (x · , z) + ( 1 /2n+ l )(y*, z) > > sup ff + x* + (1/2 n+! )y* ] - 1/4n + 1 � c � f(x) + (x*, x) + (1/2n+ 1 )(y*, x) - 1/4n + l . Using (5.9) and (5.8) we obtain f ez ) + (x * , z) + (1/2 n+ l )(y* , z) > > f ez ) + (x * , z) - 1/4n + (1/2n+ l )(y* , x) - 1/4n + 1
(5.9)
or
(1/2 n+ 1 )(y*, x - z) < 1/4n + 1/4n+ l = 5/4n+! = 5/22 n+2 . Equivalently, (y*, x - z) < 5/2n+ 1 = (5/2)(1/2 n ) = >'/2 n , which contradicts (5.7) and completes the proof. We next illustrate the power of Theorem 5.15 by using it to give a simple
proof of the geometric fact mentioned earlier, that RNP sets are generated by their strongly exposed points.
Theorem 5.20. Suppose that C C E is a nonempty bounded closed convex set with the RNP. Then C is the closed convex hull of its strongly exposed points. Moreover, the functionals which strongly expose points of C constitute a dense G6 subset of E*.
Proof. We first prove the second assertion. To this end, define
92
Section 5
Gn
= {x* E E*: diam S(x* , C, a) < l/n for some a > O } . If y* E Gn (so that diam S(x*, C, a) < l/n for some a), then by Lemma 5.18, applied to f = y*, the a/2-neighborhood of y* is contained in Gn, hence the
latter is open. Theorem 5.15 (applied to any element f of E*) trivially implies that each Gn is dense in E*, hence by the Baire category theorem, the set G = nG n is a dense Go subset of E*. Obviously, if x* strongly exposes C, then it defines slices of arbitrarily small diameter, hence is in G. Conversely, if x * E G, then it will define a nested sequence {S(x* , C, a n ) } of slices of C with diameters converging to o. Their closures intersect in a point of C which is strongly exposed by x*. To prove the first assertion of the theorem, let D be the closed convex hull of the strongly exposed points of C. If D =f C, then by the separation theorem there exists X*E E* such that O" D ( X* ) < O"c ( x *). Since these functionals are norm continuous on E* , there exists a functional in the dense set G for which the same inequality holds, contradicting the definition of D. Since strongly exposing functionals can be used to define slices of arbi trarily small diameter, the foregoing theorem leads to a nice characterization. Theorem 5.21. A
Banach space E has the RNP if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points.
The following variant (and corollary) of Theorem 5.15 due to M. Fabian [Fa2] yields a strong minimum for a lower semi continuous function f defined on a space with the RNP. It replaces the restriction that f be defined only on a bounded set with a strong lower-boundedness hypothesis for f on all of E. Corollary 5.22. (Fabian) . Suppose that the Banach space E has the RNP and that f: E --? R U {oo} is a lower semicontinuous function on E for which there exist a > 0 and b E R such that
f(x) > 2all x ll + b, x E E. Then for any E > 0 there exists x* E E* such that I l x* II a strong minimum on E.
<E
and f + x* attains
Proof. Since we can replace f by f - b, we may assume that b = O. Note that if x * E E* with I l x* II < a , then for any x E E (5.10) f( x ) + (x* , x ) � 2 all x ll - all x ll = all x ll . Let r = (l/ a )[f(O) + 1] and apply Theorem 5.15 to - f restricted to the ball B == B(O; r), with 0 < E < Thus, there exist x* E E*, IIx* II < E, and a point Xo E B such that f + x* attains a strong minimum in B at X o . We need only show that this is a strong minimum for f + x * in E. If x E E is such that f( x) + (x*, x ) :::; f( xo ) + ( x * , xo ) = infB ( f + x*) :::; f(O), then from (5.10) we conclude that I l x ll :::; ( l / ) f(O) < r, so x = X o . Similarly, if {x n } C E and a.
a
5. Asplund spaces, the RNP and perturbed optimization.
(f + x*)(x n )
--t
(f + x*)(xo), then eventually f(x n ) + (x*, x n )
from (5.10) it follows that
Xn
E B and therefore X n
--t
0, there exists x E E with I I x ll < f such that f + x attains a strict [strong] minimum on C.
Note that in this result, the function f is perturbed by a weak* continuous linear functional (which is necessarily Lipschitz). Ghoussoub and Maurey also obtain a similar result for subsets of a Banach space E which has the following property.
point of norm to weak continuity property (PCP) if every nonempty bounded subset of E admits
Definition 5.25. A Banach space is said to have the
relative weak neighborhoods of arbitrarily small norm diameter. [Equivalently, every nonempty bounded closed subset contains at least one point at which the identity map (restricted to the subset) is norm to weak continuous; this explains the terminology.] For more about the PCP, see [Ed-W] and [Gh-M1]. It is clear that the RNP implies the PCP (slices define weak neighborhoods), so the following result should be compared to Theorem 5.15. In one sense, it is more general, since it applies to nonconvex sets; on the other hand, the perturbation R need not be linear nor do we get a strong minimum; also, E is assumed to be separable. Theorem 5.26. Suppose that the separable Banach space E has the PCP and that C is a closed bounded nonempty subset of E. If f is a lower-bounded lower semicontinuous function on C and f > 0, then there exists a norm Lipschitz and weakly continuous function h, of Lipschitz constant at most f, such that f + h attains its minimum on C.
94
Section 5
Finally, N. Ghoussoub, J. Lindenstrauss and B. Maurey [G-L-M] have shown that a complex Banach space E has the "analytic" RNP if and only if for every bounded upper semi continuous real-valued function f on a closed bounded subset A of E and every E > 0, there is a plurisubharmonic function 9 on E, with supremum norm on A at most E, such that f + 9 attains a strong maximum on A. We refer to [G-L-M] for details and relevant definitions. Remarks. The fact that bounded subsets in the dual of an Asplund space admit weak* slices of arbitrarily small diameter was proved in [Na-PhJ. The converse (Theorem 5.7) was proved by C. Stegall [StelJ; the simpler proof given here is due to van D ulst and Namioka [Du-NJ. The fact that convex sets with the RNP are generated by their strongly exposed points (Theorem 5.20) has been of considerable interest. It was first proved by the author [Ph2l, using geometrical methods and assuming that the entire space had the RNP. J. Bourgain then proved the general case [BoJ, also using geometrical methods. C. A. Rogers pointed out that the original proof could easily be modified to yield the general result [La-Ph, p. 1 19J. K. Kunen and H. P. Rosenthal [Ku-RJ have proved it using vector-valued martingales, not as outlandish as it appears, if one is aware that the RNP can be characterized in terms of a martin gale convergence theorem (see [Bou]). A self-contained proof, using a Kenderov-like generic continuity theorem and the duality between differentiability and strongly exposed points, is given in [Ph3J. We have presented the result here, of course, as a rather easy corollary to Theorem 5.15. Our proof of the latter is due to J . Bourgain [Bo2 J, who used a modification of his proof that RNP sets can be characterized in terms of the so-called Bishop-Phelps property [BoIJ. It is interesting to note that Theorem 5.15 is an easy consequence of Corollary 5.22, at least in the special case when the entire space E is assumed to have the RNP. (This is the case applied, for instance, in [Cr-Lh,2])' Indeed, if f is lower semicontinuous and lower bounded (by m say) on the bounded closed convex set C and if the latter is contained in a ball of radius r > 0, then the function which equals f in C and +00 outside C satisfies the hypotheses of Corollary 5.22, taking a = 1/2 and b = m - r. Fabian's proof in [Fa2] of Corollary 5.22, apparently obtained independently of Stegall's (earlier) paper [Ste2l, uses the version of Theorem 5.20 found in [Ph2J, applied to certain subsets of E X R, and is much easier than the proof of Theorem 5.15 given here. It follows trivially from Theorem 5.20 that a Banach space with the RNP has the Krein-Milman Property (KMP): Every bounded closed convex subset of E is the closed convex hull of its extreme points. It remains an open question whether a Banach space with the KMP has the RNP. There have been a number of partial results; for instance, R. Huff and P. D. Morris (see [Bou, p. 91]) have shown that the answer is affirmative in any dual space, J. Bourgain and M. Talagrand (also see [Bou, p. 423]) have shown the same for any Banach lattice while V. Caselles [CaJ has given a short proof of this and of C .-H Chu's result [ChuJ that it is true in the predual of a von Neumann algebra. W. Schachermayer [SChlJ has given an affirmative answer for Banach spaces which are isomorphic to their squares, as well as for convex sets which are "strongly regular" [Sch2J. The reader has undoubtedly wondered whether the duality between Asplund spaces and the RNP goes the other way. It almost does: A Banach space E has
the RNP if and only if every continuous convex function on E* which is also weak* lower semicontinuous is Frechet differentiable at the points of a dense G6 subset of E* . This was proved by J . Collier [CoJ and generalized by S. Fitzpatrick [FiJ; see [BouJ for an exposition.
Section 6
6. Gateaux differentiability spaces.
The following class of spaces is formally larger than the class of weak Asplund spaces, but in some ways is a more natural object of study. Definition 6.1. A Banach space E is called a Gateaux Differentiability Space (GDS) provided the set G of points of Gateaux differentiability of a convex
continuous function defined on a nonempty open convex subset D e E is necessarily dense in D.
Clearly, a GDS differs from a weak Asplund space only by virtue of the fact that one does not require the set G to contain a dense Go . As shown in Prop. 1.25, the Go property is automatic for Frechet differentiability, but known examples (see e.g., [C-K] or [Tall ]) show that for Gateaux differen tiability (of individual convex continuous functions, at least), it is definitely an additional requirement. In this section we will examine some properties of Gateaux differentiability spaces, all but one of which remain open questions for weak Asplund spaces. Another motivation for studying Gateaux differentiability spaces is that they admit an interesting characterization which is completely analogous to Theorem 5.12 (which characterized Asplund spaces in terms of weak* strongly exposed points in their duals). We state this result now; its proof will be given below.
Banach space E is a GDS if and only if every weak* compact convex subset of E* is the weak* closed convex hull of its weak* exposed points.
Theorem 6.2. A
There is only one permanence (or stability) property known to be valid for weak Asplund spaces (they are preserved under quotient maps (Theorem 4.24)), while the Gateaux differentiability spaces have that property (even a stronger property; see Prop. 6.8 below) plus a simple but useful stability property under certain products (which is still open for weak Asplund spaces). It will help the exposition to introduce temporarily a class of spaces which is formally larger than the class of Gateaux differentiability spaces (but which will turn out to be the same).
96
Section 6
Definition 6.3. A Banach space E is said to be an M-differentiability
8pace
(MDS) provided every Minkowski gauge on E is Gateaux differentiable at the points of a dense set. Proposition 6.4. If E x R
is an MDS, then the Banach space E is a GDS.
Proof. We pair E x R with its dual E * x R by ((x* , r*), (x, r)) = (x* , x) + r * · r.
Suppose, now, that f is a continuous convex function defined on an open con vex subset D of E. We can assume without loss of generality that the origin is in D and that f(O) - 1. Let fl be the Minkowski functional defined. on E x R by the convex set epi(f); since the latter has nonempty interior [con taining, for instance, the origin], fl is necessarily continuous. If fl is Gateaux differentiable on a dense set, then by positive homogeneity it is necessarily Gateaux differentiable on a dense subset of =
G
=
((x, r) : x E D and fl(x, r) = I} .
The set G is that part of the boundary of epi(f) whose "x-coordinate" is in D, so it is the graph of f and hence it is homeomorphic with D. Thus, it suffices to show that if fl is Gateaux differentiable at the point (x, f(x)) (where x E D), then f is Gateaux differentiable at x. It is clear from Figure 6.1 below that any corners in the graph of f will be points at which epi(f) will have distinct supporting hyperplanes, hence points at which fl fails to be Gateaux differentiable.
Fig. 6.1
Specifically, suppose that xi , xi were distinct sub differentials to f at x , so that (6. 1 ) (xi , y - x) 5: f(y) - f( x ) for all y E D, i = 1, 2. Taking y 0, we see that (xi , x) - f(x) � -f(O) 1, the reciprocal S j of the quantity on the left is positive, for i 1 , 2. We claim that the two functionals s;( xi, - 1), i = 1 2 are distinct sub differentials to fl at (x, f(x)). It suffices to show that for each i, =
=
so
=
,
,
(s; (xi , - l ) , (z, r)) 5: fl (z, r) for (z, r) E E x R,
(6.2)
6. Gateaux differentiability spaces.
97
with equality at (z, r) (x, l(x)). By positive homogeneity of both sides of (6.2), it will be proved if we show that =
(S;(x i , - l ) , (z, r) ) < 1
whenever p,(z, r) < 1. The latter, of course, implies that (z, r) is interior to epi(f), so that z E D and r > I(z). Thus, by using this fact and (6.1) with y = z,
S i ( ( x i , z ) - r) < s i ( ( x i , z ) - l(z)) :S; s i ( (x i , x ) - I(x))
=
1
as was to be shown. The equality of (6.2) at (x, j(x)) is just the definition of S i , since p,(x , I(x)) 1 . Finally, the two functionals s i (x i , - 1) are distinct: if sl (xf, -1) S2 (X� , - 1), then Sl = S2 and hence xf x�. Our stability result is the following: =
=
=
Proposition 6.5.
If E is a CDS, then E x R is a CDS.
Proof. Suppose that D is an open convex subset of E x R, that I is a continuous convex function on D and that (xo , to) E D. Let U be an open neighborhood of (xo, to ); we want to find a point of Gateaux differentiability of I in U. Without loss of generality we can suppose that U contains a neighborhood of the form B x I, where B is an open ball centered at Xo and I [to - 8, to + 8] ; we can also assume that I I I is bounded in U, by M, say. Choose a differentiable extended real-valued function 9 :s; 0 on I which equals at the endpoints, is finite elsewhere and satisfies g ( to ) = O. Define =
-00
h e x)
=
sup{J(x, t) + g(t) : t E
R},
xE
B.
Since h is the supremum of a family of convex functions, it is convex; moreover, since I is bounded above by M on B, so is h. Also, for any point x E B, h( x ) ::::: I(x , to ) +g(to) ::::: - M, so h, being bounded, is continuous on B. (Recall the remark after Proposition 1 .6.) By hypothesis, h is differentiable at some point Xl E B. A compactness argument shows that h(XI ) l(x I , td + g (td for some tl E I. This implies that g (td > - 00, so that tl is an interior point of I. We will use the characterization of Gateaux differentiability as given in Exercise 1.24. For any (y, s) E E x R and all t > 0 sufficiently small, we have (Xl ± ty, tl ± ts) E B x I and =
o :s; I( Xl + ty, tl + ts) + I( Xl - ty, tl - ts) - 21( xl , tl ) :s;
h(XI + ty) + h(XI - ty) - 2h(x } ) - [ge t } + t) + g e t } - t) - 2g(td] .
Since both h a d 9 are differentiable, if we divide through these inequalities by t > 0 and let t 0, the terms on the right tend to zero, showing that I is differentiable at (Xl , tt ). n
--+
Corollary 6.6. A
Banach space E is a CDS if and only if it is an MDS.
98
Section 6
Proof. Let H be a closed hyperplane in E, so that E is isomorphic to the product H x R. IT E is an MDS, then by Prop. 6.4, H is a GDS. By Prop. 6.5, the space H x R, that is, the space E, is a GDS. 6.7. Problems. Is a closed subspace of a GDS necessarily a GDS? Is the product of two Gateaux differentiability spaces necessarily a GDS? ( It is obvious from P rop. 6.5 that, by induction, the second question has an affirmative answer for the product of a GDS wit h a finite dimensional space. )
The following permanence property for Gateaux differentiability spaces is similar to the corresponding one for weak Asplund spaces (Theorem 4.19) which required the operator T to be onto; the proof is essentially the same;
Suppose that T: E F is continuous and linear, with dense range. If the Banach space E is a GDS, then so is F.
Proposition 6.8.
---t
Proof. Suppose that D is a nonempty open convex subset of F and that I is continuous and convex on D. The function II I aT is convex and continuous on the open convex set DI = T- l (D), hence is Gateaux differentiable at the points of a dense set G 1 C DI . To see that I is Gateaux differentiable at the points of the dense set T(GJ ), it suffices to verify that T* [8I(Tx)] C 8JI(x) for all x E Dl and use the fact that density of T(E) implies that T* is one-one. =
(It follows easily from the foregoing proposition that the first question in 6.7 has an affi rmative answer for any complemented subspace M of a GDS space E; that is, for any M which is the image of a continuous linear projection on
E.)
The next proposition is competely analogous to Prop. 5.11. Proposition 6.9. Suppose that p is a Minkowski gauge on E. An element x* in C(p) = {x* E E*: (x*, x) � p(x) for all x E E} is weak* exposed by x E E if and only if p is Gateaux differentiable at x, with dp( x) = x*.
Proof. Suppose that x* E C and that x weak* exposes C at xOO; then 8p(x) {x*} . Indeed, if y * E 8p(x), then by Lemma 5.10 we have y* E C and (y*, x) p( x) = (7c ( x), hence yOO x*. Conversely, suppose that p is Gateaux differentiable at x, with xOO = dp(x). Then x* is in 8p(x), so by Lemma 5.10 again, x* E C and x attains its supremum on C at x*. Suppose there were another point y* E C such that c ( x) (yOO , x); then the other implication in Lemma 5.10 shows that y* E 8p(x), hence yOO = x*, that is, x weak* exposes C at xOO. Recall the statement of Theorem 6.2: A Banach space E is a Gateaux differentiability space if and only if every weak* compact convex subset C of E* is the weak* closed convex hull of its weak* exposed points. =
=
=
(7
=
6. Gateaux differentiability spaces.
99
Proof of necessity in Theorem 6.2. The proof that the dual of a GDS has the indicated property is identical to the proof of the analogous portion of Theo rem 5. 12, except for the use of Prop. 6.9 in place of Prop. 5.11 and, of course, the substitution of Gateaux differentiability for Frechet differentiability. To prove the converse, we need the following dual version of a classical "parallel hyperplane lemma" . Lemma 6.10. Suppose that x, y E E, I (x*, y) I :S 1 whenever x* E E* satisfies
with J l x J l
= 1
J l y ll , and
=
E
>
o.
If
(x *, x) 0 and Jl x * J I < 2jE, or J l x + y J l :S E. =
then either J l x - y J l :S E
The following sketch shows the reason for the name; the hypotheses require that hyperplanes defined by the functionals x and y be nearly parallel. y = 1
y = -1
Fig. 6.2
Proof. Note that, as a (weak* continuous ) linear functional on E*, y is bounded in absolute value by Ej2 on the intersection of the dual ball with the subspace H = {x* : (x*, x) O}. By the Hahn-Banach theorem, its restriction to H can be extended to a functional of norm at most Ej2 defined on all of E*. Since this extension is necessarily weak* continuous, it is defined by an element z E E. Thus, y - z = 0 on H, so y - z = a x for some a E R. Note that =
11
- Ial l IIl y Jl - Jl y - zJ l I :S J l zJ l :S Ej2. If a � 0, then J l x - y J l = J I (1 - a)x - zJ l :S 1 1 - al + J l zJl :S E. If a < 0, then J l x + y J l = J I ( 1 + a )x + z ll :S 1 1 + a l + II z ll :S E. =
Proof of sufficiency in Theorem 6. 2. By Cor. 6.6, it suffices to show that E is an MDS. To that end, suppose that p is a Minkowski gauge functional on E, so that, by Lemma 5.10, we can write p = 0 sufficiently small, y + t z E U and therefore T(y + t z ) C W . Applying (7.1) to any u* E T(y + tz) we get =
o �
( y * - u *, y - (y + tz)) = - t ( y * - u*, z ) ,
which implies that ( u*, z )
2':
(y*, z ) , that is, u* is not in W , a contradiction.
If T: E 2 Eo is maximal monotone and D is a nonempty open subset of D(T), then the restriction Tv of T to D is maximal monotone in D.
Corollary 7.8.
---7
105
7. A generalization of monotone operators: Useo maps.
Proof. By Example 7.2, the maximal monotonicty of T implies that the mono tone map TD is convex w*-usco, so the result follows from Lemma 7.7. The next theorem exhibits an interesting relationship between maximal monotone operators in an open set D and minimal usco maps in D.
If D is open in the Banach space E and if T: D 2 E* has nonempty values and is maximal monotone in D, then T is a minimal convex w* -usco map in D. Theorem 7.9.
-+
Proof. We know by Example 7.2 that T is convex and w* -usco, so the only question is whether it is minimal in this family. Suppose that F: D 2 E* is convex and w*-usco and that G(F) C G(T). By Lemma 7.7, F is maximal monotone and hence F = T. -+
It is easy to see that not every minimal convex w* -usco is monotone; consider any continuous non-monotonic function from R into itself. The application of these notions (usco maps, minimal usco maps, etc. ) to a generic continuity theorem for monotone operators requires several further lemmas.
Let X , Y be Hausdorff spaces and suppose that F: X 2Y has closed graph and is locally relatively compact, that is, every x E X has an open neighborhood V M£ch that F(V) is relatively compact in Y. Then F is a usco map. Lemma 7.10.
-+
Proof. Since G( F) is closed, every image F( x) is closed and contained in some relatively compact set, hence is compact. To see that F is upper semicontinu ous, it suffices to show that for each x in X and each open neighborhood V of x, the restriction of F to V is upper semicontinuous. By hypothesis, we may restrict attention to open neighborhoods V of x for which F(V) is compact. Given such a V, define FI : V 2Y by FI(x) F(V), x E V. Obviously, FI is a usco map and F l v C FI ; since G(F I �T) is closed in V x Y, Proposition 7.5 implies that F is upper semicontinuous in V. -+
=
If D is open in E and T: D 2 E* is monotone, with Iiil nonempty for each x E D, then the map T whose graph is the closure G(T) in D x (E*, w*) of G(T) is monotone and w* -usco.
Lemma 7.1 1 .
-+
Proof. Let TI : D 2 E* be a maximal monotone extension of T. By Theorem 7.9, TI is a minimal convex w*-usco map in D, and by Prop. 7.3 its graph G(TI ) is closed in D x (E*, w*). Thus, T c TI so T is monotone and-by Prop. 7.5-it is also a w* -usco map. -+
The next theorem shows how the concept of minimal usco maps can be used to prove a basic result about monotone operators.
106
Section 7
Suppose that X is a Hausdorff space and that T: X 2 E* is w* -usco. 'For x E X define coT(x) to be the weak* closed convex hull of T(x). Then the map coT is convex w* -usco.
Lemma 7.12.
-+
Proof. Since 'CoT obviously has weak* compact convex values, it suffices to prove that it is weak* upper semicontinuous. To see this, suppose x E X and that U is a weak* open subset of E* with 'CoT(x) c U. In any locally convex space, a compact convex set I< has a neighborhood base of the form I< + lV, where the closed convex sets W form a neighborhood base of o. Thus, we can assume that U is of the form U coT(x) + W, where W is a weak* closed convex neighborhood of o . By the upper semicontinuity of T there exists an open neighborhood V of x in X such that =
T(V) C coT(x) + W. It follows that coT ( V ) C coT(x) + W, so coT is upper semicontinuous. Suppose that D is an open subset of the Banach space E and that T: D 2 E* is a monotone operator, with T( x) i= 0 for all x in D. Then there is a unique maximal monotone operator M in D containing T. In fact, M can be characterized as follows: Let T be the set-valued map whose graph is the closure in D x ( E*, w* ) of G(T) and for each x E D let M(x) be the weak* closed convex hull of T(x); this defines M.
Theorem 7.13. -+
Remark. As Example 7.4 shows, one must distinguish between sets of the form T( x) and the ( possibly smaller) sets T( x ). Proof. Let TI be any maximal monotone operator containing T. By Theorem 7.9, TI is a minimal convex w*-usco map. Since it has closed graph, we must have T C T1 , and since T has closed graph, Prop. 7.5 implies that it is w*-usco. From Lemma 7.12, we conclude that the map M coT is convex w*-usco and, clearly, M C T1 • By the minimality of T1 , we have M T1 , which proves the =
=
uniqueness assertion.
There does not appear to be a unique extension theorem for monotone operators with arbitrary effective domains. If, for instance, E has dimension at least one, if (xo, Yo) E E x E* and if T is defined to be the monotone operator whose graph is {( Xo, Yo) }, then there are many maximal monotone extensions of T. The next lemma, which is purely topological in nature, has Kenderov 's Theorem 2.30 on continuity of maximal monotone mappings as an immediate corollary. The main hypothesis will seem less peculiar when we apply the lemma to maximal monotone operators.
Let F be a minimal usco map on the Baire space X with compact values in the Hausdorff space (Y, ) and let d be a metric on Y. Sup pose that for every nonempty open subset U of X there exist nonempty open Lemma 7.14.
r
7. A generalization of monotone operators: Usco maps.
107
subsets V of U such that F( V ) contains relatively open subsets of arbitrarily small d-diameter. Then there exists a dense Gli subset D of X such that F is single-valued and d-upper semicontinuous at each point of D . Proof. We first note that the fact that F is a minimal usco map implies that if J is a proper closed subset of G( F), then p( J) =f X, where p is the natural
projection of X x Y onto X . Indeed, if p( J) = X, then J would be the (closed) graph of a set-valued map which, by Prop. 7.5, would be a usco map properly contained in F. Next, given e > 0, let Of
=
U{G: G is an open subset of X and d - diam F(G) :s:
e}.
Clearly, Of is open in X j we will show that it is dense. Let U be a nonempty open subset of X. By hypothesis, there is a nonempty open subset V of U and a T-open subset W of Y such that F(V) n W =f 0 and d-diam (F( V ) n W) :s: e. Since G(F) n (V W) =f 0 and F is minimal (and, by Prop. 7.3 (a), G(F) is closed), we must have p[G(F) \ (V x W)] =f X. Choose Xo E X \ p[G(F) \ (V x W)]. Then p - t (xo ) n G(F) C V x W; that is, Xo E V and F(xo) C W. If G {x E X : F(x) C W} n V, then G is an open neighborhood of Xo with d-diam F(G) :s: d-diam(F(V) n W) :s: e. It follows that G C Of and therefore 0 =f V n Of C u n Of ' This proves that Of is dense in X. Now let -
=
D = n{O t /n: n
=
1 , 2, 3, ... }.
Since X is a Baire space, D is a dense G li subset of X . From the definition of D it is evident that not only is F( x) a singleton at each point x E D but that F is d-upper semicontinuous at each such point. This lemma leads to the following alternative proof of Theorem 2.30.
Let E be a Banach space such that every bounded nonempty subset of E* is weak* dentable. If T: E 2E* is maximal monotone, with X == int D(T) =f 0, then there exists a dense Gli subset D of X such that T is single-valued and norm-to-norm ttpper semicontinuous at each point of D . Theorem 7.15.
---t
Proof. We know from Example 7.2, Cor. 7.8 and Theorem 7.9 that T is a minimal convex w*-usco map from X into (E*, w*)j by Prop. 7.3 it contains a minimal w* -usco map Tt . By Theorem 2.28, T is locally bounded in X, hence the same is true of Tt . Consequently, given any nonempty open subset U of X there exists a nonempty open subset V of U such that Tt (V) is bounded. By the weak* dent ability hypothesis, Tt (V) admits nonempty relatively weak* open neighborhoods of arbitrarily small norm diameter, by Lemma 7.14 there exists a dense Gli subset D of X such that Tt is single-valued and norm to-norm upper semicontinuous at each point of D. Now, by Lemma 7.12, coTt is convex w*-usco, and hence by the minimality of T l x , we have coTt = T l x . so
108
Section 7
Thus, for x E X , if T1 (x) is contained in a closed ball, then so is T(x). It follows that T is also single-valued and norm-to-norm upper semicontinuous at each x E D. The following exercise exhibits a particular convex w*-usco map which is of considerable importance in optimization. 7.16. Exercises: The Clarke subdifferential. Suppose that f is a locally Lipschitzian real-valued function on a nonempty open subset D ( not necessarily convex ) of the Banach space E. For each x E D the generalized directional derivative r(x; h) of f at x in the direction h E E is defined by
r (x; h) = lim sup
( l , y )_( O + , x )
f(y + th) - f(y) . t
( a) Show that r (x; h) is finite and that for fixed x, the function h is positive homogeneous, subadditive and Lipschitzian. ( b) Show that if {xn } C D and Xn
-+
-+
r(x; h)
x E D, then for each h E E,
lim supJ" (xn ; h) :::; r (x; h). The Clarke subdifferential 8° f( x) of f at x E D is defined by
8° f(x) = {x · E E · : (x · , h} :::; r(x; h) for all h E E } .
( c ) Show that, for all x E E, the set 8° f(x) is a nonempty convex weak* compact subset of E· . ( d ) Show that the graph of 8° f is norm X weak· closed in D that 8°f is a convex weak· usco map whenever D = E.
X
E· and hence
If f is a continuous convex function, then it is locally Lipschitzian, hence one can define both its generalized directional derivative and Clarke subdifferential as above. The following proposition reassures us that these coincide with the usual notions of directional derivative and subdifferential in this case. Proposition 7.16.
If f is convex and continuous on a nonempty open convex subset D of E (hence is locally Lipschitzian), then for all x E D and h E E one has r(x; h) = d+ f(x)(h)
and hence 8° f(x) = 8f(x). Proof. It is immediate from the definitions that r( X; h) � d+f(x)(h) . To prove the reverse inequality, note that for any fixed 8 > 0, r ( x; h) = lim
sup
< - 0 + II w - " II < , 6 o < t < t:
f(y + th) - f(y )
(7.2)
7. A generalization of monotone operators: Usco maps.
109
Since (as in Lemma 1. 2 ) the difference quotient
fey + th) - f ey) t is nonincreasing as t -+ 0 + , the right side of (7.2) equals sup I·1m f - O + il y -xil < fD
fey + {h) - fey ) {
Now, there exists a neighborhood of x in which f has Lipschitz constant M say, so for all sufficiently small { > 0, if I l x - y ll < lb, then
>
0,
fey + lh) - fey ) l(x + {h) - f(x) 2bM < + { { and therefore r (x; h) ::; d+ f( x)( h) + 2b M . This being true for all b > 0, we obtain the desired inequality. The fact that the two subdifferentials coincide follows from the foregoing equality, the definition of 8° f( x) and the fact (shown in the proof of Prop. 1 .8) that 8f(x) = {x' E E' : (x' , h) ::; d+ f(x)(h) for all h E E}. Theorem 7.9 shows that if a monotone operator which takes on nonempty values is an open set D and is maximal monotone in D is necessarily a minimal convex w*-usco in D. Although the Clarke subdifferential of a Lipschitzian function f on an open set D is a convex w*-usco in D, it need not be a minimal convex w*-usco in D. Of course, it is minimal if f is convex, and J. Borwein [Bor2 ) has shown that it is minimal for certain other related classes of functions.
Remarks. Most of the results in this section were obtained independently by Drewnowski and Labuda [Dr-L) and Jokl [Jo); we have followed the exposition of [Dr-L), with some very significant improvements suggested by both S. Fitzpatrick and I. Namioka. The proof of Lemma 7.14 (which results in the new proof of one of Kenderov's theorems in Theorem 7.15) is a minor revision of one due to the highly esteemed John Rainwater [Rain), who was motivated by the work of M . E. Verona rVer). Between them, they have extended some of the generic differentiability results to Lipschitz continuous convex functions defined on a sort of quasi-interior of a closed convex set C (namely the Baire space of all non-support points of C). Some of their theorems were obtained independently by D. Noll [Nolh , 2 ). Far-reaching generalizations of these kinds of results have been obtained by J. Borwein, S. Fitzpatrick and P. Kenderov [B-F-K) and by M. E. Verona and A. Verona [V-V 1 ,2,3) . As mentioned above, the Clarke subdifferential plays an important role in opti mization; see, for instance, [Cl) for further properties and applications. While it is a very nice extension to locally Lipschitzian functions of the usual notion for convex functions, it has some drawbacks. For instance, it does not apply to lower semicontin uous convex functions which are not continuous (hence not locally Lipschitzian� , and it fails to be single-valued for some differentiable functions (such as f(x) = x sin� for x ::j:. 0, f(O) = 0; in this case it is easy to verify that 8° f(O) = [ - 1 , 1]).
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[S-S) [Tah) [Tab) [Tay) [Tr) rVer) [V-V!) [V-V2) [V-V3) [Ze)
_____
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INDEX
Absorbing point 28 Absorbing set 28 Affine function 1 a-cone 20 a-cone meager 21 Angle-small 2 1 Asplund space 1 3 , 22-25, 31-33, 79-82 Asplund space, weak 13, 33, 35, 95 Asplund's theorems 22, 35, 37, 70 p-derivative 59 p-differentiable 59 P-subdifferentiable 64 p-superdifferential 65 Banach-Mazur game 69 Bishop-Phelps theorems 44, 48, 49 Bomology 58 Borwein-0mo theorem 77 Borwein-Preiss theorem 68 Bounded, locally 4, 28 Bourgin's lecture notes 110 Br�ndsted-Rockafellar theorem 43, 48, 87 Bump function 59 Clarke subdifferential 108 Coban-Kenderov 77 Concave function 1 Convex function 1 Convex usco map 102 Cyclically monotone 26 Cyclically monotone, maximal 28, 53
Exposed point 83 Extended real-valued function 38 Extreme point 83 Fabian's theorems 66, 92 Farthest distance function 1 Fenchel transform 42 Frechet bomology 58 Frechet derivative 7 Frechet differential 7 Functional, gauge 1 Functional, Minkowski 1, 84 Functional, sublinear 1 Gateaux bornology 58 Gateaux derivative 3 Gateaux differentiability space 95 Gateaux differential 3 GDS 95 Generalized directional derivative 108 Generic differentiability 1 1 Ghoussoub-Maurey theorems 93 Hadamard bomology 58 Haydon's examples 59, 66 Indicator function 39 Inf-convolution 3 1 , 42 Infinite tree 80 Infinite o-tree 80 James' theorem 43, 61 John Rainwater 109 Kenderov's Theorems 30, 34, 107 Krein-Milman property 94 Kuo's theorem 36
o-tree 80 Dentability 24, 79 Dentability, weak* 79 Dick's lecture notes 110 Directional derivative 1 Directional derivative, generalized 108 Derivative, Frechet 7 Derivative, Gateaux 3 Distance function 1 Duality mapping 27
Lipschitzian extension 31 Lipschitzian, locally 4, 7 Locally bounded 4, 28 Locally bounded operator 28 Locally bounded subdifferential 7, 15 Locally Lipschitzian function 4, 16 Lower semicontinuous function 38
t-subdifferential 46 Effective domain, function 38 Effective domain, monotone operator 17 Effective domain, saddle function 54 Ekeland-Lebourg theorem 66 Ekeland's variational principle 45, 58, 6 1 , 88 Epigraph 38 Epi-sum 42
Maximal cyclically monotone 28, 53 Maximal monotone 26-27, 30, 56, 73, 104 Mazur's theorem 12 M-differentiability space 96 MDS 96 Metric projection 8, 18 Minimal usco map · 102 Minkowski gauge 1 , 84
116
Support point 43
Monotone, cyclically 26 Monotone, maximal 26, 56, 73, 104 Monotone operator 17, 26, 104 Monotone set 26
Talagrand's examples 77 Tree 80
N-cyclically monotone 26 Nearest point mapping 8, 18 Nonexpansive map 18
Upper semicontinuous map 18-19, 102 Upper semicontinuous function 88 Usco map 102
Parallel hyperplane lemma 99 PCP 93 Perturbed optimization 87 Positive operator 17 Preiss' theorem 68 Preiss-Phelps-Namioka theorem 73 Preiss-Zajicek theorem 22 Proper convex function 38 Proper saddle function 54-55 Property (H{J) 59
Variational inequality 8 Variational principle 45, 49, 58
Rademacher's theorem 1 1 Radon-Nikodym property 79 Right-hand derivative 2 Rockafellar's theorems 52, 53 Rosenthal's theorem 36 Rotund norm 33 Saddle function 54 Selection 19 Semicontinuous function, lower 38 Semicontinuous function, upper 88 Semicontinuous map, upper 18, 102 Simons' proof 51 Slice 24, 79 Slice, small diameter 24, 30, 31 Slice, weak* 24, 30-31 , 79 Smoot}> norm 34 Smootr space 34 Smooth variational principle, Borwein-Preiss 68 Smooth variational principle, DGZ 63 Stegall's theorem 88 Strictly convex norm 33 Strictly convex space 34 Strong maximum 88 Strong minimum 59 Strong w· H6 set 93 Strongly exposed point 83, 91 Subdifferential 6, 15, 19, 40 Subdifferential, Clarke 108 Subdifferential, saddle function 54 Sublinear functional 1 , 71 Support function 39 Supporting functional 43 -
WCG 13, 35, 66 Weak Asplund space 13, 35, 67 Weak Hadamard bornology 58 Weakly compactly generated 13, 35, 66 Weak* dentable 79 Weak* exposed point 83 Weak* slice 24, 79 Weak* strongly exposed point 83 Weak* usco map 102
INDEX OF SYMBOLS
B(x; r)
4
C(p)
84
D{3
62
df
3
d+ f
2
de
1
oe
39
D(T)
17
dom( f)
38
epi( f)
38
G(T )
26
1-
43
Pe
1
O"A
39
