Calculus in vector spaces without norm (Lecture notes in mathematics, 30)

Lecture Notes in Mathematics A collectioo of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich Series: Forschungsinstitut for Mathematik, ETH, ZUrich • Adviser: K. Chandrasekharan

30 A. FrSlicher- W. Bucher Universit~ de Geneve

Calculus in Vector Spaces without Norm 1966

•

m

Springer-Verlag- Berlin-Heidelberg. New York

Work supported in part by the Swiss National Science Foundation

AU rights, especiallythat of translation into foreign languages, reserved. It is also forbidden to reproduce this book, Other whole or in part, by photomechanicsl means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verhtg. O by Springer-Verlag Berlin. Heidelberg 1966. Library of Congress Catalog Card Numbez 66-30545. Printed in Germany. Title No. 7350

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I

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CONTENTS

IV

Introduction

1

§l. ELEMENTARY PROPERTIES OF FILTERS 1.1

Filters and filter-basis

i

1.2

Comparison of filters on the same space

2

1.3

Mappings into direct products

2

1.4

Images of filters under mappings

3

1.5

An inequality between images of filters

4

§2. PSEUDO-TOPOLOGICAL

VECTOR SPACES

6

2.1

Pseudo-topological

2.2

Continuity

8

2.3

Induced structures

8

2.4

Pseudo-topological

2.5

Quasi-bounded and equable fi3tsrs

15

2.6

Equable pseudo-topological

19

2.7

The associated locally convex topological vecfior sp~ce

21

2.8

EquBble continuity

24

2.9

Continuity with respect to the associated structures

30

§3. DIFFERENTIABILITY

spaces

6

vector spaces

12

vector spaces

AND DERIVATIVES

3.1

Remainders

3.2

Differentiability

3.3

The chain rule

3.4

The local caracter of the differentiability

32 32

at a point

35 38

§4. EXAMPLES AND SPECIAL CASES

condition

38

42

4.1

The classical case

42

4.2

Linear and bilinear maps

43

~.3

The special case

44

f: A ~ E

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4.4

§s.

§6.

II

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Differentiable mappings into a direct product

FUNDAMENTAL THEOREM OF CALCULUS

50

5.1

Formulation and proof of the main theorem

50

5.2

Remarks and special cases

58

5.3

Consequences of the fundamental theorem

60

PSEUDO-TOPOLOGIES ON SOME FUNCTION SPACES

65

6.1 6.2

The spaces B(E1;E2) , Co(El;E2) and L(E1;E2) Continuity of evaluation maps

65

6.3

Continuity of composition maps

71

6.4

Some canonical isomorphisms

72

THE CLASS OF ADMISSIBLE VECTOR SPACES

82

7.i

The admissibility conditions

82

7.2

Admissibility of E ~

84

7.3

Admissibility of subspaces, direct products and projective limits

7.4

§8.

§g.

a6

Admissibility of

6g

85 B(EI;E2) , Co(EIIE2), Lp(EIIE2)

87

PARTIAL DERIVATIVES AND DIFFERENTIABILITY

9O

8.1

Partial derivatives

8.2

A sufficient condition for (total) differentiability

go gl

HIGHER DERIVATIVES

g3

g.l

f"

g3

g.2

f(P)

and the symmetry of f"(x) for

p R 1

g5

gg

§I0. Ck-MAPPINGS 10.1 The vector space 10.2 The structure of

Ck(EI;E2) Ck(EI;E2)

gg i01

10.3 C.(E1;E2)

104

10.4 Higher order chain rule

105

-III

§ii.

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THE COMPOSITION OF Ck-mAPPINGS

ii0

ii.i

The continuity of the composition map

lid

ii.2

The d i f f e r e n t i s b i l i t y of the composition map

114

§12. DIFFERENTIABLE DEFORBIATION OF DIFFERENTIABLE mAPPINGS 12.1

The differentiability of the evmluation map

12.2

The linear homeomorphism

c.° (c1,d. (E2,E3)) - c'..(EI ,,E2~E3)

131 131 132

APPENDIX

137

NOTATIONS

14D

INDEX

143

REFERENCES

146

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IV

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INTRODUCTION

As emphasized by J. Dieudonn~, with the approximation

(in a neighborhood

calculus primarily deals of some point) of given

mappings of vector spaces by linear mappings. linear map has to be a "goad" approximation

The approximating

in some precise sense:

it has to be "tangent" to the given map. A very useful notion of "tangent" can easily be introduced

for maps between normed vector

spaces; it leads to the notion of "Fr@chet-differentiable" and gives, theory

mappings

in particular for Banach spaces, a very satisfactory

(cf. Chap. VIII of [3]).

It is well known that in this classical of differentiability

and derivative

the given norms by equivalent topologies.

theory the notions

remain unchanged

if one replaces

ones, i.e. by norms inducing the same

It is natural therefore to look for a theory which does

not use the norms, but only the topologies of the considered spaces.

vector

In fact, throwing out something which is irrelevant usually

leads to a clarification

and simplification

on one side, and allows

a more general theory on the other side. In the case of calculus, such a generalization

is indeed desirable in view of applications

to certain function spaces which have a natural topology,

but no

natural norm.

In classical theory, the norm is essentially used at two places:

(1) One defines what maps

r: EI---~E2

are tangent to zero

~t the origin (we simply ca)l them "remainders") fr~chet-oondition:

~

on the vector space

( 1 . |r (x)~

= O:

L(E1;E2) , consisting

maps from E 1 into E2, by taking,

for

by means of the

(2) One defines a norm of the continuous

~ ~ L(EI~E2):~h=

linear

~li~

(x)R.

In order to obtain a similar theory for a class of non-normed topological ~ector spaces, one has therefore to choose essentially two definitions:

(1) What are the remainders from E 1 to E2;

-V-

(2) What is the topology of L(E1;E2). The second definition comes in as soon as one wants to consider second (or higher) derivatives, since the first derivative f' of a (differentiable) map is a map

f: E1---~E2

f': E1--*L(EI;E2)o But all attempts which have been made

along this line gave theories with a very serious deficiency: the composite of twice differentiable mappings did not turn out to be twice differentiable in general; in other words: there was no higher order chain rule. In fact, a look at the classical proof shows that the second order chain rule is a consequence of the first order chain rule and of the differentiability of the composition map c: L(E1;E 2) × L(E2;E 3)

i L(E1;E3). But for non-norm~ble topological

vector spaces E i there seems to be no separmted topology on the spaces L(Ei;Ej) such that the composition becomes differentiable.(*) Nevertheless, a way out of this difficulty was found: independently A. Bastiani and H.H. Keller realized that though there is no satisfactory topology on the spaces L(Ei;Ej) , there exist pseudo-topologies which have the desired properties. The authors are very much indebted to H.H. Keller for having drawn their attention to the fact that pseudo-topologies seem really the proper thing to use at this place.

(*) This statement is not very precise, in particular since it depends on the adopted definition of "differentiable". If, ho~uever, one requires thst "differentiable" shall imply "continuous" end that the natural isomorphism between L(~;Ei) and E i shall be a homeomorphism, then one knows that with topologies one cannot succeed; in fact, the continuity of the composition map

c: L(~;E1) × L(E1;E2)

..... ,L(~;E . 2)

then is

equivalent with the continuity of the evaluation map e: L(E1;E2) × E1----oE2, and for non-normable spaces El, E 2 there is no topology on L(E1;E2) for which this evaluation map e is continuous

ET]).

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VI

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The above remarks show that it is not for the sake of greatest possible generality that we develop our theory right from the beginning for pseudo-topological

vector spaces (topo-

logical ones are special cases of these), but simply in order to obtain a satisfactory theory for a class of vector spaces containing at least some non-normable topological ones. In order to

prove certain theorems of calculus,

some restrictions however

will have to be made: a class of pseudo-topological

vector spaces,

called "admissible" ones, will be introduced. This class contains in particular ai1 separated locally convex topological vector spaces.

Since our whole theory works consistently with filters, §l starts with some well known facts concerning filters. For a reader who is familiar with filters, it will be sufficient to have a look at (1.5..2)~ we found that at some places in the literature the inequality stated there was erroneously used as an equality.

§2 presents the basic facts concerning pseudo-

topologies and in particular pseudo-tomological vector spaces. The material of sections 2.5 to 2.9 will not be used for the beginning of calculus and thus can be read later, whenever referred to.

§5 deals with what might be called the "mean value theorem". However, there is no mean value in it; but it is fundamental in the sense that it is used in order to orove ormctically all of the deeper results of calculus. We thus call it "fundamental theorem of calculus".

Intuitively,

it

gives an estimate of the difference between the endpo±nts of a motion of a point in a vector space by means of the velocity of that motion, the estimation being made by means of convex sets. In the case of normed spaces, the theorem yields the well known estimate by means of the norm (cf. (8.5.1) of [3]) provided one chooses as convex set the closed unit ball; but being able to

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VII

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take other convex sets, we get better information also in this classical case: We not only can conclude that the point does not get too far if the velocity is not too big, but also that the point does get far, if the velocity is big (in the sense of lying in a multiple of the convex set in question). some consequences in particular, inequalities

For later applications,

of the theorem are established

at the end of §5;

two versions of the theorem in the form of filter

will turn out to be useful. Another consequence

Taylor's formula;

it will be given in a forthcoming

is

publication.

In §7 one finds the definition of the admissible

spaces

and furthermore a result without which the theory would not be satisfactory:

the class of admissible vector spaces is closed under

the constructions

used in calculus,

yielding new spaces out of

given ones, such as L(E1;E2) or Ck(E1;E 2) out of E 1 and E 2. In §8 we show that the relations between partial and total differentiability

of a mapping of a direct product are as

in classical theory; in particular,

partial differentiability

continuity of the partial derivatives

plus

implies total differentiability.

We remark that this theorem uses in a very essential way the choice of the structure of the spaces L(Ei;Ej), pseudo-topologies

since "continuous"

refers to the

of the spaces in question.

The main results of §9 state that the p-th derivative at a point can be identified with a multilinear map which is symmetric, and that the composite of p-times differentiable differentiable

maps is also p-times

(p-th order chain rule).

The notion of a Ck-mapping

introduced in §lO coincides with

the usual notion of a k-times continuously the case of finite-dimensional more restrictive.

differentiable

mapping in

spaces, while in general it is slightly

The vector space consisting

E 1 into E 2 is denoted by Ck(E1;E2)

of the C~mappings

or C~(E1;E2),

from

depending on which of

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VIII

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two pseudo-topological structures we consider (we always use one symbol to denote the space and its structure). The important spaces are the spaces .C~(E1;E2) ; but for technical reasons it is useful to define them by means of the spaces Ck(E1;E2) as auxiliary spaces and a general operator "

"

which associates to any pseudo-topology

of a vector space a second one, having in addition a certain important property, called equability. In special cases, the operator " ~ " becomes the identity; in particular, if the spaces E i are finite dimensional, the pseudo-topology of Ck(E1;£ 2) = C~(E1;E2) is nothing else than the topology of uniform convergence on bounded sets of the functions and thair derivatives up to the k-th order. The cass k = ~ is obtained by forming a projective limit. In §ll the differenti~bility and the C -nature of the P composition map of Ck-mappings are investigated, the main results being theorems (11.2.21) and (11.2°26); here, the result stating that the composition map is of class C

is in fact stronger than P just saying that it is p-times continuously differentiable° §12 deals with differentiable families of differentiable

maps, "differentiable" now always meaning "differentiabls of class CN~

Having our theory of differentiation and also a pseudo-topology

on the vector space of differentiable maps from E 1 into E2, one can consider two sorts of differentiable families of such maps: a) A differentiable family of maps (depending, for instance, on a real parameter)is a differentiable map of ~ x E 1 into E2; b) A differentiable family of maps is a differentiable map of ~ into the function space C~ (E1;E2)o The main result of §12 not only says that these two notions are completely equivalent, but even that the structures put on the space of all differentiable families according to either one of the two points of view a) or b) are the same; in other words, there is a canonical linear homeomorphism between C~ (~ x El;E2) and C~ (~;C~ (El;E2)). Moreover, the "parameter space" ~ can be replaced by any admissible equable vector space E. If we consider this iso-

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IX

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morphism in the special case E = E l = E 2 = ~, for instance, the space on the left hand side is classically

then

well known, while

on the right hand side we have a new function space, consisting of functions with values in the infinite dimensional

space C.o(~;~).

Repeating this argument one sees that at least for many spaces Ei, Ej the set and the structure of C ~ ( E i ; E j )

are uniquely determined

one requires the following two conditions: are finite dimensional, functions,

if

(1) in case E i and Ej

C ~ ( E i ; E j) is the set of classical Cm-

with the topology of uniform convergence

of the functions and their derivatives;

on compact sets

(2) the linear homeomorphism

(12.2.5) mentioned before shall hold.

Depending on the choice of the two main definitions obtains different theories.

Our approach is different

of A. Basfiiani, H.H. Keller and E. Binz ([1],[6],[2]).

one

from those In order to

develop our theory, we always postulated that the definitions agree with the classical ones in the case of normed spaces, a condition which is not satisfied by the theories of A. Bastiani or of E. Binz. The structure of L(EI;E 2) defined by H.H. Keller for the case of locally convex spaces El, E 2 by means of families of semi-norms (cf. [5]), seems to be the same as the structure of our L@(E1;E2). In [6], H.H. Keller introduces various notions of differentiability and compares them with definitions other authors

that have been suggested by still

(cf. the references given there),

restricting

himself

in that paper to locally convex spaces. At the time being it is difficult to recognize which one of the various theories will eventually

turn out to be the most

useful one. That mainly depends on what theorems one gets and on what applications

one wants to make. An implicit function theorem

has not been obtained so far; in fact it is known that its classical formulation

simply fails to hold. H.H. Keller has also established

and motivated a series of basic properties that should hold in a useful theory of calculus

(cf. [7]); we believe that our theory

-X-

satisfies these conditions.

Throughout this report, we restrict ourselves to certain vector spaces; manifolds modelled on such vector spaces shall be considered later.

Though our notion of differentlability

is a local property,

a non-local condition is imposed on the so-called Ck-mappings ; but this condition becomes trivial in the case of finite dimensional spaces, and, at least, it is not so restrictive as to rule out the identity map, as it would be the case if one had to restrict oneself to maps with compact or bounded support.

The

f~r@~-named author has presented a first version of

calculus for topological vector spaces in a Seminar of Professors A. Dold and B. Eckmann at the Swiss Federal Institute of Technology (ETH), Zurich, in summer lg63; it was not yet satisfactory,

since

there was no higher order chain rule. A part of the present theory was outlined by the same author in a series of lectures at the Forschungsinstitut

for mathematik of the ETH during the 1964/65

winter term.

The present work has been partially supported by the Swiss National Science Foundation.

-1-

§ 1.

ELEMENTARY PROPERTIES OF FILTERS.

Since the whole theory is based on the convergence of filters, we recall here the fundamental facts concerning filters and state an inequality (in 1.5) which will oe used very frequently in the sequel.

i.I.

Filters and filter-basis.

A filter on a space (i.e. set) m is a non-empty set ~ o f

subsets

of m ~uch that

(1) (2) X2

(3)

X1

XI,X 2 ~ ~

~ Xl ~ X 2 ~ ~

.

A filter-basis on M is a non-empty s e t 1 ~ o f subsets of m such that

(1) (2) Sl,S 2

"B

~There exists 8 3 E I ~ with B 3 c a I n B 2 .

Each filter is a filter-basis. Conversely, each filterbasis ~determines a filter

~ = E ~ ] a s follows : ~ consists of

all subsets of M which contain a set of ~ . ~ i s

called the filter

generated by the filter-basis I~. In particular, if B is any nonempty subset of M, I~ = ~B~ is obviously a filter-basis. The filter

(*) ~always denotes the empty set

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2

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generated by it consists of all subsets of M containing the fixed set B and is denoted simply by ~B] .

Analogously, if a ~ m,

(a3

denotes the filter formed by the subsets of M containing the point ao

1.2.

Comparison of filters on the same space. The set of all filters on a given space ~I is partially

ordered by the set-theoretic inclusion~

(1.2.1)

~(I ~ X2 ~'~ ~l o 12

(*).

We thus have the notions of infimum and supremum of a family of filters on m : inf ~i and sup ~i" i~ I i~ I

The second always

exists; it is the filter consisting of all sets of the form

(1.2.2)

L~ x i , where X i ~ ~i" i~l The first does not always exist; it will not be used. According to usual notation one also writes : 1.3.

mappings into direct PrOducts. If fi : m i - ~ N i

we denote by

i~I ×

sup ( ~l' ~2 ) = E1 V ~2'

' i~ I

resp. i = 1,2, are mappings,

>~ m i the direct product of the sets Mi~ and by i~I

fi ~ i~I x mi__iXiN i resp. flxf2 : mlXM 2 --~ NlXN 2 the maps

defined as follows:

(*) In [4] Fischer uses the symbol "~ " in the other sense; definition (1.2.1) is the one used e.go by Kowalsky in[ ~] .

-3,,

(1.3.1)

resp. iEl

iEl

i~l

(flxf2)(Xl,X2)

In the special case where M. = ~ for all i~I, X m .

=

(fl(Xl),f2(x2))o

is usually

denoted by m I, and we further denote by

T[f.

:im

-'--'~ ~N.

i61 i

i~l

reap.

Ill,f2] : m ~

NIXN 2

i

the maps defined as follows:

(i .3.2)

(~fi)(x) iGl

= ~fi(x)}

respo

[fl,f2] (x) = (fl(x),f2(x)).

i¢I

These maps are related by means of the diagonal maps d : m

jmI

respo

d : m

~ mxm = m 2

as follows :

(1.3.3)

f. : ( X f i ) , d iEIi iGI

resp. [fl,f2] : (flxf2).d.

1.4. Imaqes of filters under mappings. Let f: m ~f(X) I X E I ~ i s

~N be a mapping and ~ a filter on ~. The set

than a filter-basis on N, which generates a filter

denoted by f ( ~ )

and called the image o f ~ u n d e r

the mapping f.

The use of the same symboi f is justified because the functor in question is covariant :

(1.4.1)

If M

f~ N

g :P, then (g.f)(~) = g(f(~));

and also because

(1.4.2)

f([x3 ) = ff(x)].

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4

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The induced mapping for filters is order-preserving

(i.4.3)

Xl ~ ~ Let now ~

(1.4.4)

'

'

:

f(~l)~ f(~)"

be a filter on mi, i = 1,2. Then we define:

~l x ~

is the filter on m I x ~2 generated by

fx xx l If further g : m l X M 2 - - * N

(i.4.5)

g(~,

is a map, then:

~2) denotes the filter generated by the

?o~oo~n~ ~er

~,~X~,X~) I X~.~X~,

~)

It folloms easily that

(i.4.6)

o(I i, ~2) = o(~ I x X2).

1.5. An i neguality between imaqes of filters. Let X be a filter on m and d : m ~ M x ~ 1

be the diagonal

map : d(x) = (x,x). Then

(i.s.i) Proof. Let A~](x~. we have: A ~ X x X ,

Then A ~ X l X X 2, where Xl,X2~l(.

where X m X .

But since XxX~d(X),

With X = Xl4X 2

it follows that

A m d(X) which shows that A cd(~(). Usually, this inequality will be used in combination with mappings.

(i.5.2)

A typical example is as follows.

Let h i : m --,Ni, i = 1,2 and g : NlXN 2 - - ~ p If f : M

be maps.

; P is the map defined by f(x) = g(hl(X),h2(x)),

then f(K) ~ g(hl(@) , h2(~())"

-

Proof.

S i n c e by ( 1 . 3 . 3 )

5

-

we have f - ge [ h l , h 2 ~

= g,(hlXh2).d

the inequality foliows from (1.5.i), using (1.4.3), the equality (hlxh2) ( @ x ~ )

= h1(~)

,

(1.4.6) end

x h2(~).

We shall refer to (1.5.2) whenever we have occasion to use an inequality of this type, even though the situation may be somewhat different.

For example it could be that there are several

variablesp or the right hand side might contain x repeated more than once.

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§ 2. .

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PSEUDO-TOPOLOGICAL .

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6

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VECTOR SPACES. .

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For a detailed introduction of the notion of pseudotopological spaces, the reader is referred to [ 4 ]

. We shall

introduce a slightly different notation which will be more convenient for our purpose. introduced,

Some associated structures will be

and questions concerning continuity will be discussed.

It turned out that a condition which is slightly stronger than ordinary continuity will play an important role: the notion of equable continuity.

In particular we investigate the case

of linear and multilinear maps.

2.1.

Pseudo-topological

spaces.

A pseudo-topology

(or limit-structure)

on a space

(i.e. set) M consists in assigning to each x¢ ~ a set of filters on m, such filters being described as "converging to x". The following axioms are supposed satisfied: to x, then so does any smaller filter; to x, then so does their supremum;

(1) If a filter converges

(2) If two filters converge

(3) The filter Ix] converges

to x. A pseudo-topological

space

E

consists of a set

together with a pseudo-topology on m. The set m is called the

-7

-

underlying space and will be denoted by ~.

If a f i l t e r ~ o n

converges to x in the sense of the given pseudo-topology, write

~x

E (~converges

we

to x on E). The axioms can now be

expressed in the following way.

(2.1.1)

For all

(21 ~

(3)

x E E

~x E

[,3

for

one has :

i = 1,2

---@ t l v

t 2 ~x E ;

E.

Topological

spaces can be c o n s i d e r e d as s p e c i a l

topological

spaces.

define

In f a c t ,

if

cases o f pseudo-

E is a topological

space, we

:

X~ x E where ~ x

~ x

is the filter

verging filters s i n c e then

~

'

o f neighborhoods o f x . Knowing t h e con-

on E, c o n v e r s e l y we can r e c o v e r the t o p o l o g y ,

~ . . = sup

~

. Thus, a necessary c o n d i t i o n

for a

X @xE pseudo-topological

(

sup ~ ) ~x E.

space t o be a t o p o l o g i c a l

one i s t h e F o l l o w i n g :

This condition is not in general sufficient:

~x E however we shall see in (2.4) that it is sufficient in the case in which we shall be interested,

i.e. if E is a pseudo-topological

vector space. For details, see [4] .

-8-

Continuity.

2.2.

Let El, E 2 be pseudo-topological spaces and f: ~i--~p~2 a map.

We say that f: EI---pE 2 is continuous at the point

a( E 1 iff (*)

El

f(~) ~f(a) E2 "

f: EI---$E 2 continuous means: continuous at each point. In the topoIogical case the definition is equivalent to the usual one. One further verifies easiIy that the composite of continuous maps is also continuous. f: E I - - ~ E 2 is a homeomorphism means that f: ~ i - - - ~ 2

is bijective

and that f: EI---~E 2 as well as f-l: E2._.~EI are continuous.

2.3.

Induced structures. Une can introduce a partial ordering on the set of

pseudo-topologies of a fixed space.

Let El, E 2 be two pseudo-

topoiogical spaces with --El = ~2"

(2.3.1)

Definition: The structure of E1 is called finer than that of E2, and we write E 1 ~ E2, iff

~x El ~

~ ~x E2"

(*) From time to time we shall make usa of the expression "iff"; as usual it stands for "if and only if".

.

g

,.,,

We also say the structure of E 2 is coarser than that of El . This is equivalent to the condition: the identity map i : E1----PE 2 is continuous. As in the topological case, one can define structures which are induced by mappings. We will use only the following case: Let Ei, i~ I, be pseudo-topological spaces, M a space (without structure), and fi : m ...... ; ~i

, i~ I .

Then there exists on ~ a unique pseudo-topology which is caracterized as being the coarsest pseudo-topology on m such that all maps fi : ~-'--~Ei are continuous.

Denoting m together

with that structure by E, we have (@ being a filter on m) : (2.3.2)

X& x E ~

fi(X) &fi(x) E i

for all i~ I.

In Fact, one easily verifies that with this definition the axioms (2.1.1) are satisfied; and (2.3.2) certainly defines the coarsest structure on m such that all maps fi : m " - * E i get continuous. If all the spaces E i have the property that for each point xE E i one has

sup ~ ~x Ei ' then also E has this property. i~xE i

This is easily verified, using that for any map f : M ~ any family

IXj~ j 6J

of filters on M, one has

N and

-

(2.3.3)

f ( sup

j~J

~j)=

sup

j~3

lO

-

f(~j)

;

which i s an immediate consequence of the s e t - t h e o r e t i c

f (jV xj)

f(xj)

equality

.

We consider now three s p e c i a l cases of s t r u c t u r e s induced by mappings : subspaces, d i r e c t products and p r o j e c t i v e limits. a)

Each subset A of a pseudo-topological

pseudo-topology, i : A ~

E o

subspace of E.

namely the one induced by the inclusion map A together with this structure is called a Denoting the subspace thus obtained by E 1 ,

we have according to (2.3°2)

(2.3.4)

space E has a natural

~ x El ~ b) Given a family

:

i (~)$x E.

~Ei~ i~l of pseudo-topological

call direct product and denote by space whose underlying

spaces, we

X E i the peeudo-topological iGI

set is the direct product of the sets _Ei,

together with the structure induced by the projections "~k : X _Ei ~ i6I

Ek ,

ke I.

Thus (2.3.2) yields for this case:

(~.3.5)

&x (× Ei) ~

ITk (~) ~ ~k(X) Ek for all k~.

i~ I In case of finite direct products we write E 1 x...x E n instead of

X E. . i e (l,...,n~ z

-

c)

l l

-

Let a projective system of pseudo-topological

i.e. to each element i of a directed set II,~} gical space Ei is associated, subset of ~i 2 ,

--

E

i6I

a pseudo-topolo-

~uch that for i I ~ i2, ~i I is a

with a continuous inclusion map.

is the pseudo-topological section E = F ]

spaces be given,

Then L = proj.limi( I Ez

space whose underlying set ~ is the inter-

and whose pseudo-topology

is the one induced by

--z

the inclusion maps Ji : ~ --'-* El" In the case of subspaces and direct products, the following two~lemmas will be used later.

(2.3.6)

Lemma.

Let f : EI----~E 2 be a map and suppose that E 2

is a subspace of E3, i : E 2 ---~E 3 being the inclusion map.

Then f : E1----~ E 2 is continuous if and only if

ief : EI----~ E 3 is continuous. Proof.

Necessity is obvious, since i : E 2 ----, E 3 is continuous.

Suppose conversely that i-f is continuous, (iof)(~) $(iof)(a)E3,

or equivalently

By (2.3.4) this yields f(~) ~f(a)

and let

[~a

El" Then

(by (1.4.1)): i(f(~))$f(a)E 3.

E2' which proves the continuity

of f.

(2.3.7)

Lemma. Let X there exist

ZlX) 2.

be a filter on ElXE 2. Then X~xEIXE 2 ~l' ~ 2

with ~i &%i(x)Ei ' i = 1,2, and

-

Proof.

12

-

i ) Since for any subset X© ElXE2 one has Xcg(X)x l~'2(X),

one has for any filter ~ on ElXE2:

(2.3.8)

~Tl(X ) x ~2(~) • Hence, if ~ x E l X E 2 , we can choose ~i = TFi(~)" 2) If, conversely, ~ ~ ~l x ~2' where ~i ~ ~Ti(x) Ei' then ~i(~) ~ ~i ( ~ l x @ 2 ) = Xi, and according to (2.3.5) this p~oves that ~ x E l X E 2 .

2.4. Pseudo-topological vector spaces. Let the underlying space ~ of a pseudo-topological space E be a vector space (*). The pseudo-topology is called compatible with the vector space structure (shortly: compatible) if the maps ExE ~xE are continuous.

+ "

~E, ~E

By IR we always denote the reals, taken with the

usual topology. A pseudo-topoloqical vector space is a vector space together with a compatible pseudo-topology on it. more precisely: it is a triple, consisting of a set together with two structures on it, namely an algebraic structure (of vector space) and a pseudo-topological structure.

(*) Vector space here always means: vector space over the reals ~.

,- 1 3

-

Continuity of addition implies that the translations are homeomorphisms. Therefore: (2.4.1)

~a

E ~-~

~-a &o E

~+ a ~a E ~ = ~ o

By map

or equivalently

E-

~-a we denote of course the image of ~ under the translation x ~--*x-a.

g(x,y) = x-y.

More generally, we write g ( ~ , ~ )

=~-~

if

Then one easily verifies that ~ - a can also be

considered as the image of two filters under the difference map: -a = ~ -

[4

° Similarly for ~ + a : ~ + a =~,[a].

In view of (2.4.1) the pseudo-topology of a pseudotopological vector space E is completely known if we know what filters converge to zero.

Hence we only need one relation, that

of "converging to zero", and we shall simplify the notation by writing

~ b E instead of ~ o

E. Thus " 6°.." simply means :

"converges to zero in...". The continuity of the operations implies the following compatibility conditions: (2.4.2)

(1)

"~I~E , "~2 ~ E

.---m ~'l ~ ' ~ E ~

(2)

~ ~E , ,-)$.IR

~

~1"3['J, E;

(3)

~ ~E

~

|I/."j[ ~I,E;

(4)

x mE

~

IV, x ~- E.

-

14

-

By W we always denote the filter of neighborhoods of OG IR. For the meaning of

~l + ~2' ~ '~' |V,~ and | V . x ,

see the remark

following (2.4.1) above. Conversely, if for a given vector space we say what filters converge to zero, and if the set of these filters not only satisfies the conditions (2.1.1) for x = O, but also the above compatibility conditions (2.4.2), then we obtain, taking (2.4.1) as definition, a unique compatible pseudo-topology on E. Looking at the induced structures studied in section 2.3 for the case of pseudo-topological vector spaces, one easily verifies that for linear maps fi the induced pseudo-topology is also compatible. We therefore have En particular:

Vector

eubspaces, direct products and projective limits of pseudotopological vector spaces are also pseudo-topological vector

spaces. Suppose now that for a pseudo-topological vector space E also the following condition holds:

(2.4.3)

s u p X ~E. By (2.4.1) we have then more generally: ~x

= sup ~E ×

and hence

~

x

E,

-

15

-

Let U ( 1.Lx. Then U = x + V, where V • t t o. Since by (2.4.2) we have % c

rb~° + ~ o '

there exists V'( ~ o with V ~ V '

Hence, with U' = x ÷ V' • ~x' we have U ~ U '

+ V'.

+ V' ~ y + V I •

Y

for all y( U'. We thus have shown:

for each U e ~ x there exists

U'( % x

Or it is well known that this

with U 6 % y

for all y EU'.

implies that the filters respect to a topology.

(2.4.4)

Proposition.

1&x are the neighborhood

This yields the following Condition

2.5.

and equable filters.

On pseudo-topological the property

IV,~dE

quasi-bounded

filters on E.

(2.5.1)

vector spaces,

will be frequently used.

filters • with We call them

The name is motivated by the

result. Lemma.

On a normed vector space E, a filter is

quasi-bounded Proof.

vector space is

vector space.

qMasi-b£unded

following

(cf[4 ] ):

(2.4.3) is necessary and

sufficient in order that a pseudo-topological a topological

filters with

Let | V . ~ E .

if and only if it contains a bounded set.

Hence

hood filter of zero in E. i.e. V : Ix EEI

|xl~

iV.~

~'Tv~', where I~ is the neighbor-

If V denotes the unit ball in E,

1] , then V ¢ ~ ,

hence a f o r t i o r i

V ( I~. ~

o

-

16

-

Each set of~V contains a closed interval where

~0,

and so there exists

In particular,

if x E B ,

~0

then ~

If U s ~ =

E/~

, choose

~ ~0

one has IG

and therefore

~V. ~

Conversely, ~ £1R with

~I 4 ~

hence U ¢ ~V ~

for

suppose I@ contains a

such that x ~ U for

.BcU, ~

and B • I~ such that V ~ I s ' B.

'xjI&l, which yields ~x~I ~ i/~

x ~ B, showing that B is bounded. bounded set B; so there exists

I~ = { ~ s IR i J~I ~ ~

for all x • B.

Uxll ~

~

° Taking

, showing that ~

~IV"

E.

A filter ~ on a vector space is called an equable filter iff it has the property any topological

~V. ~

= ~

. It is well known that on

vector space the filter of neighborhoods

of

zero is an equable filter.

(2.5.2)

Lemma.

A filter ~

on a vector space is equable

if and only if it satisfies the following conditions: (1) Each X ~ ]~

(2) X¢.'~

contains an X' ~ ]~ with 11.X'~I(*)

, ~.1/.0 ~

~; .X e X .

Proof. i) Suppose ~V.~ = ~ . Let X s ~ o Hence X E ~ . ]~, and thus there exist ~ ~ 0 and X 1 s ~ we have X' ~ X shown, X ~ X '

and hence

with X ~ I ~

and I1. X' = X'.

= IiX~, we get:

~X £ " ~ .

. X 1. Taking X' = I s' X l,

Let further

$ ~0.

Since, as

--

2)

Suppose, conversely,

If X e ~

l?

-

that conditions

(1) and (2) hold.

, it follows that X ~ X' = fIX', where X t~ ~ , and

since IIX' £ ~2.~( we have X 6 W" ~( • And if A G ~ V . ~ ~ then there exist

~ ~0 and X 2 ¢ X

and since ~X 2 6 ~(

with A ~ I $

by condition

X 2, hence A ~ ~X 2,

(2) we have A c ~ , which completes

the proof.

(~.5.3)

L emma. Proof.

X I and ~(2 equable ==~ @l + ~(2 equable.

For arbitrary subsets Xl, X 2 of F one has

I~ • (XI + X 2) c Is X l + 1 6x2; one easily verifies that under the hypothesis liX I = XI, llX 2 = X 2 one gets equality: l& (XI +X 2) = I6 X I + Is X 2. Let now A ~ W A > I ~ . ( X I + X2) , where

~>0

and X i •

.(~( I + ~ 2 ). Then

](i" Since

)~i is

equable, we can according to (2.5.2) choose X i such that X i = I1. Xi, so that by the above equality we get: A ~ I ~ which shows that A ~ ~/.l(1 + •. @ 2 "

XI + 16 X2,

We therefore have:

\v. ( ~l + ~2 ) ~- \v. ~l + W. ~2" Combining this with the converse inequality

W'( ~(1 + X 2 ) ~ ' @ l

which is true for arbitrary filters, since it follows from (xI + x 2) = ~ x I + ~ x 2 by (1.5.2), we get

W , (~i +~2 ) = ~V.~l + W . ~ l "

~l + ~2"

+ W.~,

-18.

(2.5.4)

Lemm_,___~a.Let ~i resp. Then Proof.

~l and

~2 be filters on E1 resp. E2.

~2 equable =====~ ~l x ~2 equable.

As before, one has for arbitrary subsets X1 rasp. X 2

of E1 resp. E2 :

I~, (Xz x X2)c I~ Xl x I% X2, while for subsets satisfying IiX i = X i one has equality. Therefore one gets for arbitrary filters

(2.5.5)

W. ( ~ I x ~E2)& ~v.XI x W . ~ 2 , while for equable filters one gets as before also the converse inequality, and hence

W (2.5.6)

(~i x ~2 ) = W ~l x~v

2 = ~i x ~2"

L=emma. Let '}El' ~2 be filters on E. Then ~l and

){2 equable ==-dp ~l

~/ ~2 equable.

Proof. From the set-theoretic equality 18 ,(X l v

X2) = I~

X1 u I & . X 2

one deduces, using (1.2.2), that for arbitrary filters "~l' onE:

(2.5.7)

\v.(X I ~ ~2) =~v. ~lV~V. ~2" From this, the lemma follows at once.

-

2.6.

lg

-

Equable pseudo-topological vector spaces. A pseudo-topological vector space E is called equable

iff for each ~ with ~ i

with

~

E there exists an equable filter ~ ~

E; i.e. iff

(2.6.1)

---,

It follows from the remark preceding lemma (2.5.2), that each topological vector space is equable. However, not all pseudotopological vector spaces are equable (*). Given any pseudo-topological vector space E, we can introduce on ~

a new pseudo-topology, thus obtaining a new

pseudo-topological vector space E I~" . It is defined as follows:

(2.6.2)

(1) (2)

E~

= ~ ;

~ ~ El~'iff there exists ~

with ~ =

~" ~

E.

One has to verify that the set of filters caracterized by the above condition (2) satisfies the conditions (2.1.1) (for x = O) and the compatibility conditions (2.4.2). Of the conditions (2.1.1), we only verify the second one, the others being obvious.

So let

~i~ E~

for i = 1,2.

Hence there exist

filters ~ i with

~ ~- ~i =w.

~ E

, i = 1,2.

(*) Examples will be given in a forthcoming publication.

-

20

-

Using lemma (2.5.6) we get therefore v

Here, ~l V

+

']

~2 ~ E since E satisfies the conditions (2.l.1), and

we see therefore that

~l ~ @ 2 ~ E+k"

We next verify the compatibility conditions. (i)

Let

~i~ E+ for i = 1~2. As before we getp this time using

lemma (2.5.3) :

~z + ~ which shows that

(2)

Lst

~l + X2~EI~' since ~l + ~2 ~

E.

II~E $ , and ~=IR. Then

and thus

which shows that ~ + ~ E ~, since

~ % ~ 4 E.

(3) and (4) follow immediately, since the equality (W'V)

N

: ~'~

~ (~.~)

implies that each filter of the form

~.~

: is

equable. And according to the definition (2.6.2), each filter which is equable and converges to zero in E also converges to zero in E +

. This completes the verifications.

We next remark, that E ~

is clearly an equable pseudo-

topological vector space and that (cf.(2.3.1)) always

(2.6.3)

E~

zt- E,

with equality if and only if E is equable.

-

(2.6.4)

Lemma. Proof.

E l ~ X E2~

I) Let X ~ E l m

21

-

: (E1 x E2 )~ .

x E2@

. Hence, for i : 1,2:

IT.(~z )~ Ei~ , which implies "~.(~)z_z -- ~ i =W ~ i

~ El'

Using (2.3.7) and (2.3.6) we get :

This shows that

~ ~ (E1 x E2 )~

, since by (2.5.4) ~1 x ~2 is

equable.

2) Let ~ ~ (E1 x E2)e. Hence ~ _z.~ = \V ~ ~E 1 x E2. From this we get ]Ti(~ ) ~ -Ei(~) = ]Ti(~V~) = ~V. ~ i ( ~ ) which implies I~(z X) ~ Eir

2.7.

~ Ei ,

(i= 1,2). Therefore ~ ~ El~, xE2~

•

The associate Q locally convex topological vector space. For any filter ~ on a vector space, we denote by ~ or (X)'~

the filter generated by the convex sets of ~ ;these in fact form a filter basis, since the intersection of two convex sets of is again a convex set of ~

(2.7.1)

A ~

~

~

. We thus have:

there exists X e i( , X convex, X C A.

We further define

(2.7.2) From the definitions it is obvious that one has

(2.7.3)

X

°

-

(2.?o4)

Lemma.

22

-

A set A belongs to ~ o

a set X • ~

if and only if A contains

which is convex and satisfies

where ~o,1J is the closed unit interval of

[O,1] • X = X, IR.

Essentially the proof of this lemma only uses the fact that if a set X is convex and contains the point O, then it satisfies Let now E be any pseudo-topological

~O,l~ , X = X.

vector space. We

introduce the following filters on E :

(2.7.5)

r~L = sup ~E

~

;

A

(2.?.6)

tY= I L = ~L °

The equality ~ = ~ o thus

(27.7)

~

~/~0]

= ~

Lemma.

a) b)

Proof.

holds since ~0] ~ E

implies ~

~[0] and

. The filter ~ K h a s

the following properties:

[o] _--..'1/'~ ~V'[x] z_ ~

c)

"~ 1~ ~ ~

d)

W t# ~IY:

a) Since [0] ~E,

for all x ¢ E for all ~

[0] ~ sup

~

= ~

;

IR ;

furth.r

~

~

by (2.7.3). b) Same argument, using that

~V. Ix] ~ E.

c) For ~ = O, this is a). For ~ ~ 0 ~

it follows, since

E if and only if 3 - ~ ~ E and since V is convex if and only

if ~.V is convex.

-

d) Let V e ~ ' . U ~I ~ -~=~,

and thus -U e ~ ,

and satisfies I l ~ /

we get

-

Then, by (2.7.4),

V

cnntains a set

which is convex and satisfies U = [0,~ . U. By c),

V ~ V n(-V) ~ U ~ (-U) ( ~

I~I ~ l

23

and x e U

since U E I~ c~

. Thus we have

. The set ~/= U ~ ( - U )

= ~/. In fact, if z 6 I l . ~

~(-U), i.e. x ( U

and - x ( U o

is convex

, then z = ~x, where Since ~ O , ~ . U = U,

~x = I ~ l (~x) @ U. Thus we have V ~ I l ~ / ,

which shows

that V e ~ V ~ o e) Let V 6 1 ~ , also ½U e ~ ;

and choose U as before. By (c) we have

and since ½U is also convex it follows that ½U ~ .

Using again the convexity of U we have: V ~ U ~ ½ U

+ ½U ~ + ~

,

which shows that V ( I ~ + ~ o We define now on ~ a new structure, and denote ~ together with this new structure by E° :

(2.7.3)

X

°

-

>

°

Lemma (2°7°7) immediately implies that (2°7.8) defines a compatible pseudo-topology on ~ (cfo (2.1.1)) and (2.4.2)). This pseudo-topology is in fact a topology, as follows from (2.4.4) or simply using the well known fact that the conditions of lemma (2°7.7) are necessary and sufficient in order that ~ i s

the neighborhood-filter of zero

for a unique compatible topology on ~ (cf. [Q] ). Moreover, since has, by definition, a basis consisting of convex sets, we have:

-

(2.7.9)

Proposition.

24

-

For any pseudo-topological vector space E,

the space E° defined above is a locally convex topological vector space. We f u r t h e r

remark t h a t

as a consequence o f

(2.7.3)

one

has (2.7.10)

E ~

E° ,

with equality if and only if E is itself a topological locally convex vector space.

2.8.

Equabl~ continuity. f : E1----P E 2 being any map between pseudo-topological

vector spaces, we denote by ~f:

E1 x E1 ---~E 2 the map

defined

by

(2.8.1)

~f(a,h) = f(a + h) - f(a).

(2.8.2)

Definition.

l (2.8.3)

f: E1 ~ E

2 is called equably c o n t i ~

iff

J

Proposition.

If f: E 1 ~

E 2 is equably continuous, then

it is continuous (i.e. continuous at each point a ~ El). Proof. get

Let ~ a

El" Then, since

A f( Ca] , ~ _ Fa] ) vI E 2.

we get, using (1.5.2):

~V.[a] $ E 1 and

~-[a]

~ E l, we

But since Z~f(b,x-b) = f(x)-f(b)

4 f ( #a] , X

-[a])~

f(~)

- f([a]).

-

Therefore f ( [ ) the c o n t i n u i t y

- f(a)~E2,

25

i.e.

-

f([)

~f(a)E2,

which proves

of f at a.

The n o t i o n of equable c o n t i n u i t y c o n v e n i e n t l y w i t h respect to composition,

does not behave the composite of

equably continuous maps not n e c e s s a r i l y being equably continuous. However we w i l l

have t h i s convenient behaviour i f

we add a

supplementary c o n d i t i o n . (2.8.4)

Definition.

A map f : E1

bounded map i f f

it

sends quasi-bounded f i l t e r s

quasi-bounded f i l t e r s ,

~#l ~E 1 ~

~.~ E2 i s c a l l e d a quasi.-

i.e.

into

iff

V ' f ( ~ ) ~ E 2.

We denote by ~

(El;E2) the space of equably continuous

and quasi-bounded maps of E1 into E2. C_o (El;E2) is of course closed under addition and multiplication by scalars and is therefore a vector space.

(2.8.5)

Proposition.

If feC_o(E1;E 2) and g~C_o(E2;E3) , then

g . f E C_~(E1;E3).

Proof. From th~ definition of the operator~ it follows that (&(g,f))(a,h) = Z3g(f(a),Z3f(a,h)). Hence we get by (1.5.2):

z f( ,x )).

-

If we assume that ~

~and

26

-

~,it

follows from the assumptions

made on f and g that the filter on the right side of this inequality converges to zero in E3, hence also

3

which proves that gof is equably continuous. that gof is quasi-bounded; hypothesis (2.8.6)

but that follows at once from the

that f and g are quasi-bounded

Proposition.

It remains to show

(using of course

If E 1 is a finite dimensional

(1.4.1)).

vector space

with its natural topology and E 2 a normed vector space, then C~(E1;E2)

consists

exactly of the continuous maps

from E 1 into E 2. Proof.

We already know ((2.8.3)) that, even for arbitrary El,E2,

the elements of --oC(El;E2) are continuous. Let now~ conversely, a quasi-bounded

filter~on

set. Its closure,

f: E1---JPE2 be continuous. E 1 . By (2.5.1), ~ c o n t a i n s

Consider a bounded

which we denote by A, is then a set of ~

is closed and bounded.

which

BuG such a set in a finite dimensional

vector space is comp~act,

f being continuous,

f(A) is compact. Hence f ( ~ ) by (2.5.1) this implies that

we conclude that

contains the bounded set f(A), and f(~)

is a quasi-bounded

We have thus shown that f is a quasi-bounded show that f is equably continuous.

So

filter.

map. It remains to

let, as before, W ~

~ E1 ,

-

and further

~E

27

-

1. We choose again a compact set A ~ ~

in addition a compact neighborhood V of zero in E1 . ~ i

, and denotes

the neighborhood filter of zero in Ei. The set A + V is compact (since A and V are), and hence the continuous map f is uniformly continuous on A + V. U1 ~

~l

This means: for every U 2 ~ ~ 2

there exists

such that f(y) - f(z)G U 2 for all y, z • A + V with

y-z • UI. We can choose U1 sufficiently small, such that U1 c V, and then me have in particular: all a ~ A ,

x~U1,

~f(a,x) = f(a+x) - f(a) G U 2 for

i.e.:Z3f(A,U1)¢U 2. Therefore U 2 ~ z ~ f ( J ~ , ~ l ) .

Since U 2 6 ~ 2 was arbitrary, we have ~ f ( ~ , ~ J C l ) ~ ~

~i'

we have a f o r t i o r i ~ f ( ~ , ~ ) ~

~2'

~2"

And since

i.e. ~ f ( J ~ , X ) ~

E2.

This establishes the equable continuity of f. Proposition. If ~ : E1---~E 2 is linear and continuous at

(2.8.7)

the origin, then ~ is quasi-bounded and equably continuous,

Proof.

By the linearity of ~ : ~.~(x)

we

= ~ (~.x)

and

~

(a,h) = ~ (h),

get .

and

=

and hence the continuity at the point zero yields the two assertions.

-

28

-

We denote by L (£1;£2) the vector space formed by the l i n e a r continuous maps from E1 to E2. The above lemma therefore says: L_ (Ei;E2)c C_o(q;£2) By (2.8.7) and (2.8.3), as in the topological case, the continuity of a linear map at zero implies the continuity at each point.

Howevsr~ for bilinsar (and multilinsar) maps, the situa-

tion is different: continuity at the origin does not necessarily imply continuity at all points.

(2.8.8)

Lemma.

A bilinear map b: E1 x E2 ~

E3 is equably

continuous if and only if it satisfies

i E2

b(ll,A2)~£3

(2)wail12 ~E2Ei ) ~====9b(Ai'~2)~E3 proof, (2.8.9)

we use that for a bilinear map one has ~b((al,a2),(hl,h2)) = b(al,h 2) + b(hl,a 2) + b(hl,h2).

(1)

In order to prove first that the given conditions are

necessary, suppose b equably continuous and let Ill ~£1; |V A2 ~ £2" We put ~ = [0] x J~2; ]( = ll x GO]. Then, using (2.5.5) and

(2.3.7), we have ~V'~ i El x F2 and

~ i E1 x F2. Hence by the

equable continuity of b: ~ b ( ~ , i ) ~

£3. But this is the f i r s t

condition, since for our choice of ~ and X one h a s A b ( ~ , ~ ) b( )~I'~ 2 )"

=

-29-

The second condition is verified similarly. (2) Let us now suppose, conversely, that the two conditions are satisfied, and let ~V. ~JE I x E2, ~ E filters ~ i = T i ( ~ ) '

1 x E2. Then the

]£i = ~i (K) satisfy~V.~ i ~ E i , ~i ~ El'

i = 1,2. Using (2.3.8), (2.8.9) and (1.5.2) we get:

~b(~,~) 6 ~b(~l, ~2' ~i' ~2 )~b(~l' ~2 )+b( ~I' ~2 )+b(xl' ~2 )" By assumption each of the 3 filters on the right Converges to zero on E3. Therefore also their sum and afortiori the left side, which proves that b is equably continuous.

(2.8.1o)

Lemm__~a. If the spaces El, E2 are equable~ then a bilinsar map b: E1 x E 2 ~

E3 is equably continuous

if and only if it is continuous at the origin. Proof. Necessity of the condition follows from (2.8.3). In order to prove sufficiency, let

~ I ~ E I and

E1 being equable, there exists ~ i with ~i ~

~V. ~ 2 ~ E 2 " ~i = W N I ~

El"

Thus

b(~(l,d12)~,b(W~l,~ 2) : b(~l,kV~ 2) = b(~l~V~2). But by (2.3.8), ~ i x ~V d~2 ~F1 x E2, and from the continuity of b at the origin we obtain b ( ~ l , Jl2)~E3~e thus have shown that the f i r s t of the conditions of lemma (2.8,8) is satisfied; the second one is verified in the same way, and the result follows from the lemma.(2.8.8).

- 30 -

(2.8.i1)

Proposition.

If b: E1 x E2-_..~E 3 is bilinear and

continuous at the origin, then b is quasi-bounded. Proof.

Let W . ~ E

1 x E2. As before, J~i ='r.(~l ) satisfy

~'~i~ Ei' i = 1,2 and ~ ~ ~I x~2. Using that ~ = W.W and ~-/~b(Xl,X 2) = b(~Xl,/~ x2) we get

W b(a)~.b(@ll,

J~2) = ~ . ~ . b ( ~ l , ~2 ) = b(~VJll'W~2 )'

and by the continuity of b at the o r i g i n , the f i l t e r

on the~

r i g h t and hence also the one on the l e f t converges to zero on E3.

2.9.

(2.9.1)

Continuity with respect tp the associated_structures. Proposition. If a linear map [ : E1---~E2 is continuous, then also ~" El~---~E2 ~ and ~ : E1 ° _--~E 2o are continuous. •

Proof. (1) Let ~ E 1

~ o Hence ~ ~ = ~ E 1 ,

t(~)~(~)

=~ (~V~) = W - [ ( ~ ) ~E 2,

~E2~.

which shows that ~ (~) (2) Let ~ i

and we get

= ~st~ Ei ~

, i = 1,2• By (2.3.3) for any

continuous mapping f : E1----~ E 2 with f(O) = 0 we have f(~l)

: f('sup

)( ) : sup f ( ~ )

l

(.sup ~ =~/~2"

LE1

But by the linearity of ~ we get also ~ (${i °) which ends the proof•

=

({(~i)

)o

~ 2 °,

-

(2.9.2)

31.,

Proposition. If b: E1 x E2 ~

E3 is bilinear and

continuous at the origin, then b : El~" x E2~---~E3@" is continuous (and hence equably continuous by (2.6.4) and (2.8.10)). Proof.

According to (2.B.lO) it is sufficient to show that

b: El~ x E 2 ~ E 3 ~ • i~Ei ~ , i = 1,2.

is continuous at the point (0~0). So let Hence ~i ~ ~ i = ~ V ~ i ~ E i , and

b(~l,~2)~-b(~l,~2 ) = b(W~l,~2) shows that b(~l, @2 ) ~ E 3 ~ "

=~V b ( N l , ~ 2 ) ~E 3, which

-

§ 3.

32

-

DIFFERENTIABILITY AND DERIVATIVES.

In this section, the definition of differentiability is given and the most elementary results of calculus are proved.

3.1.

Remainders. Let r : E1 ~

E 2 be a mapping between pseudo-topolo-

gical vector spaces El, E2. In order to formulate the condition which will replace the classical condition of Fr~chet, we associate to r s new map

~r: IRxEI - - ~

~r(~,x) =

(3.l.l)

Ilo/~'r(~x)

E 2 defined by

if ~4"0' if ~ = 0 .

(3.1.2)

Definition.

r: E1 ~

we write r • R(EI;E2)

E 2 is called a remainder, and iff r (0) = 0 and

w ~ ~ E1 =-~ er( W, ~ ) ~ E2.

(3.1.3)

Proposition. r: Proof. of

Let

E1~ ~ ](~EI~.

If

r : E1 4 E

2 i s a remainder, then

E2~I i s continuous at the p o i n t zero. Thus

~ ~, ~ = ~ E

~ r and since r(O) = 0 we have r ( ~ . x )

l,

By the d e f i n i t i o n

=~.~r(~,x),

and hence

using (1.5.2). r ( ~ ) ~ r ( ~ ) = r( ~ - ~ ) ~ V ~ ( W , ~ ) . Here the filter ~ = , . ~ r ( ~ V , ~ )

satisfies } -- ~V.I~ ~E 2 and there-

fore r(~) ~E2~" , which completes the proof.

-

33

-

As corollary we remark that for equable spaces El, E2 every remainder is continuous at the origin.

(3.l.~)

Proposition.

R(EI;E2) is a vector space, i.e. if

ri 6 R(EI;E 2) and Qi G ~ for i= 1~2 then

Xz rz * ~2 r2c~(Ez;C2)" Proof.

It is sufficient to show (l)

r i ~ R(EI;E2) ~=~ r i + r 2 ~ R(E1;E2)

(2)

r I R(EI;E2) , ~ ~ R ====~r = R(EI;E 2)

(i) From the definition (3.1.i) of the operatmr •

it is obvious

that ~)(r 1 + r 2) (~,x) = ~ r l ( ~ , x ) + e r 2 ( ~ , x ). Let now ~ / . ~ E 1. Then we get, using (1.5.2)= e ( r l + r2) ( ~ / , ~ ) ~ = ~ ) r l ( W , ~ )

+ ~)r2(~],l~)

and since each term on the r i g h t converges to zero~ the l e f t side does also. (2) is a special case of the following result.

(3.1.5)

L emma.

r @ R(E1;E2) ~ E L_(E2;E3) Proof.

==~

I~-r 6 R(EI;E 3)

)

The linearity of ~ implies

(e(L.r))

~,x) = ~ (erC~,x)).

-

34

-

Thus (@(~er))(W,

Ib) = ~ ( ~ ) r ( W , ~ ) ) .

Supposing now that ~V.~E1, we have 8r(~/, ~ ) ~ E2 by the assumption made on r~ and hence, using the c o n t i n u i t y of (at the origin) : [( @r( ~V,'~)) ~E3~ which completes the proof.

(3.1.6)

Uemma. r12 e R(Ez;E2) ~ ~(EI~E 2)

r2~ R(E2~E;) Proof.

I

r : r23o(r12+ ~) ~ R(EI~E3).

From the definition (3.1.i) of the operator e and the

linearity of ~ one obtains

: (e (r23o(e Thus by (1.5.2): er(~V,~) ¢~ er23 (W,~(I~) +@r12 (%V,I~)) - er23(\v,~), where we put ~ = ~

(~) ÷ @r12 (~1,1~), which satisfies

~v~ ~ ~v[(~) +~v @rz2 ( W , ~ ) If we assume now that ~ V ~

El, we see that ~V~E2,hence

@rl2(~V,~) ~E 3, and the above inequality for ~r(IV,'~) shows that also

e r ( W , ~ ) ~ E3, which proves that r is a remainder. The next result is only valid if the secono space

is separated.

-

C3.1.7)

35

-

Definition Ccf~4] ) A pseudo-topological

space E is

separated iff it satisfies

~ × E and

X~yE --~ × = y.

For a pseudo-topological vector space E this implies in particular ~ E

(3.1.8)

~

x= 0

(*)

Lemma. If E 2 is separatedp then the only remainder r : E1 ~ Proof.

E 2 which is linear is the zero map.

Let x ~ E 1 . By (2.4.2), ~V. [x]~E1, hence

But since for a linear map r one has that

~r~,x)

@ r ( ~ V , [ x ] ) ~ E 2.

= r(x), it follows

~r( IV, Ix] ) = r(~x]) = ~r(x)] . So we have ~r(x)] ~ E2,

hence r(x) = O.

3.2.

Differentiability at a point. In order to make use of (3.1.8) we assume henceforth

that all spaces El~ E2,... are separated. Let f: E1----~ E 2 be any map of pseudo-topological vector spaces, and a ~ E1 .

(3.2.1)

Proposition.

There exists at most one ~eL(E1;E2)

such that the map r defined by f(a+h) = f(a) + ~(h) + r(h) is a remainder.

(*) This condition is in fact sufficient to make E separated;

of. C4].

-

Proof.

36

Suppose there exist two, ~

-

and e2, such that the maps

r i are remainders, where ri(h) = f(e+h) - f(a) - ~i(h),

i = 1,2.

Then the map r = rI - r2 = ~2 " ~l is by (3.1.4) a remainder and is linear, hence is zero by (3.1.8), which completes the proof.

(5.2.2)

Definition.

If there exists a

~¢

~(E1;E2) such that

the map r defined by f(a+h) = f(a) +

~(h) + r(h)

is a remainder, then the map f: El

~ E 2 is said to

be differentiable at the point a; and the map

e ¢ ~(E1;E2)

which by (3.2.1) is uniquely determined, is then called the derivative of f at the point a. It will be denoted as follows: ~ = Df(a)

C3.2.31

or

~= f'(a).

Example: A constant map K: E1-----~E 2 is differentiable at each point

a G E1 •

It will be convenient to write Df(a).h instead of (Df(a))(h). If f is differentiable at the point a, the uniqueness of the derivative implies the uniqueness of the remainder r.

-

37

-

We use a similar notation :

(3.2.4)

r = Rf(a).

I (3.2.5)

Therefore:

t(h) = Rf(a)(h), r(~h)

and

= (8(Rf(a)))(~,h)

Propositio n .

If f: E1

point a, then f: E1$ Proof.

~

=~Rf(a),(A,h) E 2 is differentiable at the

~ E2¢ is continuous at the point a.

By assumption, f(a+h) = f(a) + [(h) + r(h), where

~(E1;E2)

and r6 R(E1;E2). By (3.1.3), r: El4F-- ~ E2~ is conti-

nuous at the point zero. Further, ~: E~----~ E2b is continuous according to (2.9.1). Now the result is obvious. Remark.

If in definition

(3.2.2) one only requires

linear, one still gets the uniqueness of the derivative, as the proof of (3.1.8) shows. However, it is essential for the theory, to impose the condition that ~:E1----+E 2 is continuous, since otherwise the proof of the chain rule would not work. On the other side, we see no reason to restrict the considerations to mappings f: E1----~E 2 which are continuous;

in view of (3.2.3),

what we need is that f : El~----~E2 ~ is continuous.

For equable

and in particular for topological spaces, this distinction disappears,

since then Ei = Ei~ .

-

3.5

(3.3.1)

38

-

The chain rule. Theorem.

g f ~ E 2 ___--~E 3. Then :

Suppose we have maps E1

f differentiable at a ~ E1 I ~=~Ig,f differentiable at a, g differentiable at b=f(a~

ID(g.f)(a)=Dg(b) • Df(a)

Proof. By assumption one has f(a+h) = f(a) + ~l(h) + rl(h)

g(b+k) = g(b) + ~2(k) + r 2 ( k ) , where tI = Df(a)@~(E1;E2),

rI • R(EI;E2),

~2 = Dg(b) ~ ~(E2;E3) ,

r2 G R(E2;E3).

Composing the.mappings one gets, using the linearity of ~2 : (g,f)(a) = (gof)(a) ÷ ~(h) + r(h),

where

~(h) = ~2(~l(h)) and

r(h) = ~2(rl(h)) + r2(~l(h)+rl(h))

Obviously, ~= ~2° ~l ~ ~(E1;E3)" Further, using (3.1.5), (3.1.6) and (3.1.4), re R(E1;E3) , which completes the proof.

3.4

The local caracter of the differentiabilit~ condition. Since we consider pseudo-topological

spaces E, we have

to say what we mean by "local". We call E-neighborhood of a point x ~ E a set U with U6sup " ~ = ~ x ,

which means: U •

-

39

-

for all ~ with ~ ~x E. By (2.7.3)p each neighborhood of x ~ Ee is an E-neighborhood of x. A set which is an E-neighborhood of each of its points is called an E-open set. In particular, each set which is open in E° is an E-open set. In [ 4 ] Fischer showed that for arbitrary pseudotopological spaces, the E-open sets are the open sets with respect to a certain topology (p. 273). In general, the filter ~x

is strictly finer than the neighborhood filter of x with

respect to the mentioned topology. Fischer states that in the case of pseudo-topological vector spaces (and more generally for pseudo-topological groups) equality holds and he establishes this by showing that ~ o + ~ o

= ~o

(cf'E4 ] ' Satz 6, p. 294).

Since it seems to us that there is a gap in his proof, we do not use the above topology (not knowing whether it is compatible), but only the associated locally convex structure introduced in 2.7.

(3.4.1)

Propgsition.

Suppose that two maps fi: E I " ' ~ E 2

(i = 1,2) coincide in an El-neighborhood U of the point a~ E1 . Then if fl is differentiable at the point a, f2 is also and f~(a) = f½(a).

o

Proof.

40

-

By assumption we have, for ell h G E 1 : fl(a+h) = fl(a) + ~l(h) + rl(h),

where ~I~L(E1;E2)

and rl¢ R(EI;E2).

We define r 2 : E1 -----~E2 by f2(a+h) = f2(a) + el(h) + r2(h). The proposition is proved, if we show that r2~ R(EI;E2). Since we have fl(x) = f2(x) for xG U, we get: r2(h ) = rl(h )

for all

h ~ U - a =W.

Here, by (2.4.1), W is an El-neighborhood of O, which means W~ ~

for all~

with ~ E I.

Let now ~ be a quasi-bounded filter on El: @ @ ~ E1 • Hence W~.I~,

so there exist V ~ W and B ~

with V.BcW. One has

therefore: @rl(~.x ) = er2(~,x )

for

~ ~ V, x ~ B.

But this implies that erl(~V,I~) = @r2(\V,I~). A ~ @rl(~V,~),

In fact, if

then there exist V16 W and Bl~1~

A ~ @ rl(V1,B l) ~ @ r l ( V l n V , B l ~ B ) that A ~ @r2(~V,~).

with

= @ r 2 ( V l n V,Bl~ B), showing

Similarly fore the converse. But now it

is obvious that rI ~ R(E1;E 2) implies r2GR(E1;E2).

-

41

-

Because of the local caracter of the differentiabiiity condition it would be appropriate to consider maps defined on E-open subsets of the respective pseudo-topological vector

spaces E. For a map f :

A ----~E2,

where A ¢ E1 i s an El-open

s a t , one would i n t r o d u c e d i f f e r e n t i a b i l i t y

at a p o i n t a6 A

and f'(@) by choosing any extension ~ of f to El, e.g. by t a k i n g ~ ( x ) = 0 f o r x ~A. However, i n order to s i m p l i f y presentation,

the

we s h a l l c o n t i n u e to consider maps defined on

the whole space.

-42-

§ 4.

4.1.

EXAMPLES AND SPECIAL CASES.

The classical case.

(4.1.i)

Proposition.

If El, E 2 are normed vector spaces, on

which we consider the pseudo-topology

(i.e. topology)

determined by the norm, then the notions of differentiability at a point and derivative of a map f: E1---~E 2 coincide

with the classical notions in the sense of

rr~chet (cf. [3 ]). Proof.

All we have to show is that a map r: E1----~E 2 is a

remainder (i.e. r(R(E1;E2) , cf. definition

(3.1.2)) if and only

if it satisfies the classical Fr~chet condition

(4.1.2)

lim ×-~o

Ur(x) il

llxII

(1) Suppose r satisfies

q(×) =~II

(4.1.3)

= O.

(4.1o2), and let W'B~E 1. We put

llr(x)ll if x÷0,

L"

ifx~O.

Then, by (4.1.2), q is continuous at the point O, and thus q(W'~)~IR,

which means: q(~V.~)~V.

Since one has II~r(~,x)ll = ~xlI' q(~x), one gets using (1.5.2):

II er ( w, S )11 a, II S II • q (w.a)

= U q( ~. ~)" ~

II

II V. 1BII.

But since ~.a~E 1 and the norm is continooos, II~'~IlJJR, hance also e r ( W,S)II ~ ~, whioh yields, by the definition of the pssodotopology induced by the norm: er( W , ~ ) ~ r& R(E1 ; E2) .

E1 • This shows that

-

(2)

43

-

Suppose conversely that r~ R(E1;E2) and let q be as before.

It is sufficient to show that q is continuous at the point Oo So let ~ E l ;

we shall show that then q(~)~IR. We still introduce

the map s: El ~ E

l by I

s(x)

=

l

. x

for x ~ O ,

Itxlt 0

for x =

O.

Since for all x6 E1 , lls(x)~l~l, s ( ~ ) certainly contains a bounded set, hence by (2.5.1) is a ~ o u n d e d

II ~11 ~

4.2. (4.2.1)

w

and q ( x )

= ~

@r(Uxll,s(x))ll

filter on E1 . We further have

•

so ~e get,

using (1.s.2):

Linear an d bilinear maps. Pro ppsition.

A linear and continuous map f: E1 ~

E2

is differentiable at each point a ¢ E1 and f' (a) = f. The proof is obvious: taking e = f and r = 0 one has f(a+h) = f(a) + ~ ( h ) +

(4.2.2)

Lemma.

r(h), and ~ L ( E 1 ; E 2 ) ~ rER(E1;E2).

Let b: E1 x E 2 ~

E3 be bilinear and continuous

at the point zero. Then b~ R(EIxE2;E3). Proof.

One has for all ~61R, (x1,x2)~ ElXE 2 : 8b(~,(Xl,X2)) =~.b(Xl,X2).

Hence

~b(\V,~)

- ~V.b(~).

Further, since ~.~.b(Xl,X2) = b(aXl,/~x2) , one has \V.b(~l, ~2 ) = ~" \V.b(~l,~2 ) = b(~V~l, ~V~2).

-

44

-

Suppose now that~.'B ~E 1 x E 2. One has 1 ~ 6 T l ( ~ ) where

~'l = ]T.(~)I satisfies W ~ i ~ E i ,

eb(W,I~) (4.2.3)

=~.b(~)~.b(

i = 1,2.

x IT2cB), Hence

e l , q~2) = b ( ~ V ~ ,

~1~2)1E 3.

Proposition. Let b: E1 x E2 ----~E 3 be b i l i n e a r and continuous. Then b is differsntiable at each point a = (al,a2)~ E1 x E2, and b'(al,a2)(hl,h 2) = b(hl,a 2) + b(al,h2). Proof. One has, for h = (hl,h2)E E 1 x E 2 : b(a+h) = b(al+hl,a2+h2) = b(al,a2) + b(hl,a2) + b(al,h2) + b(hl,h2). So we have, with ~(h) = [(hl,h2) = b(hl,a2) + b(al,h2) and r(h) = r(hl,h2) = b(hl,h2)s b(a+h) = b(a) + ~(h) + r ( h ) .

is obviously linear, since b is bilinear, and also continuous, since b is continuous. Thus {C~(ElXE2;E3). And r~ R(ElXE2;E 3) by the preceding lemma. This completes the proof. @

4.3.

(4.3.1)

The special case f: IR

• E.

Proposition. If f: ~R ~ E

is differentiable at the

point ~ {~, then the following limit, which we denote by f'(o(), exists:

(4.3.2)

f'(~) Further one

= lim

then has:

f(~ +~)-f(~)

.

-

Proof.

45

We have f((+~) = f(~) + ~(X) + r(~), where ~¢L(IR;E)

and r GR(IR;E). We put [(1) = f'(~).l = a and define q: IR---~ E by q(~)

~f(~÷~) - f(~)

if~,O,

~

~f ~ = 0.

One has

e(~) = [(~.i) = %. t (i) = ~ . a . Hence r(~)

Since q(~)

-- ¢ ( , ~ . ~ )

er(~,~)

:

Or(~,l)

= q(~)

- f(,~)

f(~+~)

W.I~IR we have

k

- ~.a,

" f(~)

-~(.a

- a. (Definition

3.1.2)@r(~V,1)~, E,

hence

- a~ E resp. q( IV)~a E, and that means exactly that

lim q(~) = a ; the proof is complete. ~--~o In the classical case, the converse of this proposition holds. However it seems, that in the general case, our differentiability condition is a little bit stronger. Again some question of equability comes in.

(4.3.3)

Pro~osltion.

If the scalar multiplicationIRxE

~ E

of E is equably continuous (thus by (2.8.10) in patti.(*) cular if E is equable or even topological), then the existence of the differential quotient (4.3.2) is sufficient for the differentiability of f: IR the point ~ . (*) more generally if E is admissible (see appendix (i)).

rE at

-

Proof. Let a = lim ~--~o

r

(w)

46 -

, q being defined as before; we

can write therefore: q( W)-a ~ E. We define ~: fR---* E by : ~(~) = ~.a, and r: IR--~E by r(~) = f~+~) - f(k) . ~(~). Obviously ~ ( I R ; E ) ,

and it only

remains to prove that r~R(IR;E). We havep according to the above de£1nition of r: ~r(~,W)

= ~.(q(k~)-a).

Thus we get, if ~ ,

i.e. if W ~ ~ ~, using (1.5.2):

@r(~,fl@) ~ ~. (q( W ~ ) - a ) ~ ~ . (q(W)-B). On the r i g h t , we have the product of a quasi-bounded f i l t e r onlR with a zero-converging

filter

on E, which converges to zero under

the hypothesis that scalar m u l t i p l i c a t i o n i s equably continuous.

4.4.

(4.4.1)

Differentiable mappings into a direct product. Proposition.

Let fi : F i

~ El, i ~ l , be a family of

mappings of pseudo-topological vector spaces. Then (cf ( i . 3 . i ) ) X f i : ~ < F i - - - ~ X Ei iel ilI i~l is differentiable at the point (xi)i~ I if and only if fi : Fi i~I,

Ei is differentiable at the point x i for all and then ( X fi)'([xi) i~ I ) = ~ f'(xi)° i6I iQI i

-

47

-

In particular:

(flxf2)'(Xl,X2)

=

(q(Xl), f (x2))

The proof is a combination of the following two lemmas. (4.4.2)

Lemma.

Let ~i : Fi

) Ei, i~ I, be maps. Then

X ~ i ~ L ( X Fi; . M E i) ~ icl i~I i~I all il I .

~iaL(Fi;Ei ) for

Proof.

One verifies separately that >~ [i is linear if and ieI only if each ~i is linear and that ~ ~. is continuous if and o n l y i f each ~.1 i s c o n t i n u o u s .

(4.4.3)

Lemma. Let ri : F i

~ Ei, itl, be maps. Then

Mr ~ R(X Fi;XEi) ~ iEI i i~l i~l Proof. Let us denote, for j~I, by 3

ri~ R(Fi;Ei) for all itI : XEi i~I

~E. and J

wj

:XFi---'~FjI the projections, and put r = M r . . ie iGI I One has (4.4.4)

r.

J

ow

.

J

=

a) Let r j 6 R ( F j ; E j ) follows that

"~.

J

or,

for a l l j * I .

From (3.1,6) and (4.4.4) i t

Ir..r6R(X Fi;Ej) for all jeI. Suppose now that J i~ I

WI~ ~ X F.. Then e(1"jor)(W,I~) ~Ej for all j6I. But "~. being i~I i j linear we have ~(Tj.r) = ~'jo@r. Hence Irj(~r( ~V,~))~ Ej for all jeI, which yields

@r(~V,I~) ~ X Ei. This shows that r~[~(>¢ F i ; X El). il I iG I i~ I

-

b)

48

-

Let r ER(Xi~Ifi ;i¢~Ei)I . Then, by (3.1.5) end (4.4.4) we get

rj.,wj IR(~Fi;Ej)i¢l for all jtl. Denoting by ~j: fj ----~>~i61Fi the map caracterized by the conditions

wk -~j . 0 for k ~-j,

wj o~fj = identity, we now conclude from the continuity and linearity of

Lfj by (3.1.6) that rj = rj,(wjo~fj) = (rj,wj)Q~j

belongs to R(Fj;Ej).

(4.4.5)

Proposition. Let fi : E----~E i, i61~ be a family of mappings of pseudo-topological vector spaces. Then (cf.(1.3.2))

~T i~I

f. : E i

~ ~ Ei is differentieble iel

at the point x~E if and only if fi : £ ----~Ei is differentiable at the point x for all iEI, and then:

( IT f i ) , (x) . "it f: (x). i~I i~I In particular:

Efl, f2],(x> ° [q(x>, Proof. a) Let all the maps f1 be differentiable at x. Then by the preceding proposition ~ fi is differentiable at d(x), where d is i~l the diagonal map of E into its 1-fold direct product El. Further, the map d: E ~

E l being linear and continuous, is differentiable

by (4.2.1). Now the differentiability of -~'fi and also the given iEl formula for its derivative follow from the chain rule, (1.3.3) and

(4.2.1).

-

b) Let

49

-

Jl f be differentiable. Since f =IT.o II fi' where i~l x J J i~l

the projection map 7. is linear and continuous, the differenJ tiability of fo follows again from the chain rule and (4.2.1). J

-

65

-

§ 6.

PSEUDO-TOPOLOGIES ON SOmE FUNCTION SPACES.

6.1.

The spaces B(E1;E2), Co(El;E2) and L(EI;E2). The set _B(EI;E2) of all quasi-bounded maps from E1 into E2

(cf.(2.8.4)) is a vector space, since from %V.(fl+f2)(~) & W . f l ( ~ )

+ ~V.f2(@) (cf. (1.5.2)) and

~V.(Af) (%) = A.(~V.f(%))

one deduces easily that any linear com-

bination Al.f I + ~2~f2 of two quasi-bounded maps fl' f2 is also quasi-bounded. We denote by B(E1;E2) this vector space together with the pseudo-topology caracterized by the following condition: (6.1.1)

¢~8(E1;E 2)

Iff

~V.B~El= ~

~(I~)~E 2.

@e claim that this definition yields a compatible pseudotopology on the vector space _B(E1;E2). For this, we have to verify that conditions (2.1.1) are satisfied at the point zero, and that the compatibility conditions (2.4.2) hold.

Of the three condi-

tions (2.1.1), only the second one, which demands that ~l v ~2~B(EI;E2)

if ri ~B(EI;E2) for i = 1,2, is not obvious.

But this follows easily, making use of the equality

(6.1.2)

(FlVr2)(

)

= rl(

)

which is a consequence of the set-theoretic equality (FIWF2)(B) = FI(B) w F2(B).

-

66

-

The compatibility conditions (2.4.2) follow respectively from Ca)

(rl + r 2 ) ( ~ ) ~

(b)

(~.r)(~) = ~.~(~)

(c)

(w.r) (~) = v.Y(~)

(d)

(W.[f])(~)

=

rl(~ ) + r2(~ )

(cf.(i.5.2));

W.f(1~) .

We remark that for the verification of the fourth compatibility condition it is essential that B(EI;E2) is not the space consisting of all mappings from E1 into E2, but only of the quasibounded ones. We next consider the aubspace consisting of the quasibounded and equably continuous maps from E1 into E2, which was already introduced and denoted by C.Co(EI;E2) in section 2.8. We denote now by Co(El;E2) the vector space Co(E1;E 2) together with the pseudo-topology induced by the inclusion. We thus have:

(6.1.3)

F~Co(EI~E2)

iff

V . ~ E1 --~ r(1~) ~ E2.

In the same way, L(EI;E2) denotes the pseudo-topological vector space whose underlying space is the space ~(E1;E 2) of continuous linear maps from E1 into E2, together with the pseudotopology induced by the inclusion of ~(E1;E 2) in Co(El;E2) or in B(EI;E2). Therefore we have:

(6.1.~)

r~ LCEI~EZ)

iff

~.~ ~ El ~

¢(~) ~ E2 .

-

67

-

We shall also need multilinear maps. Let ~(E!,...,En;E~ be the vector space consisting of the equably continuous multilinear maps from EIX...×E n into E. We have ~(E1,...,En;E ) c~(EIX...XEn;E),

and

thus we can consider on ~(EI,...,En;E) the pseudo-topology induced by that of B(ElX...×En;E ). Together with this structure we denote the space of equably continuous multilinear maps by L(E1,...,En;E), and we have therefore:

(6.1.5)

~ * L(EI,...,En;E )

v. ~i I E i

iff

for i = i,...,o ~

The case E 1 = E 2 . . . .

F(~l,-..,~n)

~ E.

= E n will be of special interest

and we will use the shorter notation

(~.l.6)

Ln(E1;E) = L(EI,...,E1;E). Besides these structures we can consider the structures which are associated to them by means of the operators " ~'' or ,,.,,. It will be convenient not to write the operator ~t the end, but immediately after the B,Co,L or Ln; e.g.

(~.I.?)

Co$ (E1;E2) = (Co(El;E2))~ In special cases, the structure of B(EI;E2) can be topological:

(~.l.B)

Proposition.

If E l is a normed vector space snd E 2 a

topological one, then the pseudo-topology of B(EI;E2) is the topology of uniform convergence on bounded sets.

-

Proof.

68

-

Denoting by B*(EI;E2) the space B(E1;E 2) together with

the topology of uniform convergence on bounded sets, we use that this topology is caracterized as follows:

F~B*(E1;E 2) if and

only if for each bounded subset B of El,

~(B) ~ E 2. Using lemma

(2.5.1), one shows immediately that

r ~ B.(q ~E2) ~

f ~ e(q ~E2),

which completes the proof. Combining this result with Proposition (2.8.6) we get now immediately

(6.1.9)

Proposition.

If E1 is a finite dimensional vector

space with its natural topology and E2 a normed vector space, then Co(El;E2) is the space of continuous maps from E1 into E 2 with the topology of uniform convergence on bounded sets.

(6.1.10)

Proposition.

If El and E 2 are normed vector spaces,

then L(E1;E2) is the space of continuous linear maps from [~ into E2 together with the topology induced by the usual norm (cf [ 3]) on ~(E1;E2). Proof. One combines (6.1.8) with the well known fact that the normtopology on ~(E1;E2) is the topology of uniform convergence on bounded sets.

-

(6.i.n)

Remark.

69

-

However, if we assume E1 and E2 to be topolo-

gical, not even L(E1;E2) is topological in general. In fact, as it was shown by H.H.Keller (cf.[5 ] ), there does not exist for non-normed vector spaces a topology having the properties which we shall need and which we shall verify in the sequel. The structure used in our theory on L(E1;E2) is different from the one used by Bastiani [I ] and Binz ~2 ] . In the case of locally convex topological vector spaces, our structure is related to a pseudotopology introduced by H.H.Keller in [5 ] , using families of seminorms, as follows: the structure of Keller is that of our L f (El;E2). A detailed discussion will be given in m forthcoming paper.

6.2.

(6.2.l)

Continuity of evaluation maps. Proposition. The evaluation map e: Co(El;E2) x £I ~

E2'

defined by e(f,x) = f(x), is continuous. Proof.

We show continuity at (f,x). So let ~ ~f Co(£1;£2) and

X ~x £1" Then

~-

f~Co(E1;E2) and %/. X ~E I. By (1.5.2) we get

TC ) Cf-

+

-.

"]'0

-

The first term on the right side converges to zero on E2, the second (using that f is continuous) afortiori

to f(x); hence the sum and

the left hand side converge to f(x): ~ ( ~ )

(x) E2 '

which completes the proof. We remark, that we did not use the equable continuity of f here; but continuity was essential.

(6.2.2)

Corollary. The following evaluation maps are conti~UOUS:

e: L(EI,...,En;E ) x E I x ... x E n e: Lp(E1;E ) x E1 x ... x E 1 ~

• E; E.

Since evaluation of multilinear maps is a multilinear mapping, we get by the generalisation of (2.g.2) to multilinear maps:

(6.2.3)

Proposition.

The evaluation map

e: Lp ~ (El;E) x El$ x ... x El$-------~ E ~ is continuous. The evaluation map of (6.2.1) is not, however, bilinear. But since (2.5.3) implies Co~ (E1;E 2) x E1 ~ Co(El;E2) x El, we get"

(6.2.4)

Proposition.

The evaluation map

e: Co~ (El;E2) x El is continuous.

~ E2

-

71

-

One of the difficulties of the theory is due to the fact that these evaluation maps, such as e.g. the map e: L(EI;E2) x EI---gE2,

6.3.

(6.3.l)

fail to be equably continuous in general.

Continuity of composition maps. Proposition. The composition map

c: B(El;E2) x L(E2;E3)---~B(E1;E3), defined by c(f,t) = ~°f, is continuous. Proof. Let ~ ~gB(EI;E2) and£ ~kL(E2;E3). The map c being bilinear, we have, for f, g¢ B(EI;E2) and ~, k•L(E2;E3): ~.f-

k,g : ( ~ - k ) . ( f - g )

+ k,(f-g)

+ (t-k).g.

This i m p l i e s , using ( 1 . 5 . 2 ) : c(~ ,~) - c ( g , k ) ~ (~=k).(~-g) Let n o w ~ . ~ E

+ k'(~-g)

* (~-k),g.

1. Then, again using (1.5.2):

(c(~,Z)-c(g,k))(~)~(~-k)(~

-g)(~)+k((~-g)(~))+(~

-k)(g(~)).

Since each of the three terms on the r i g h t hand side converges to zero on E3~ also the left hand side does. This implies:

- c(g,k)

B(EI

E3),

which proves the contiffuity of c at the point (g,k).

-

72

-

Using (2.3.6) we get:

(6.3.2)

Cqrollary.

The following composition maps are conti-

nuoue:

c$

L(EI,...,En;E)xL(E;F)

c=

Lp(E1;E2)xL(E2;E3 )

L(EI,...,En;F);

; Lp(E1;E3).

Again, these composition maps fail to be equably continuous in general. However, (6.3.2) implies by (2.g.2) the following

(6.3.3)

Proposition. The following composition maps are equably continuous: c: L ~(EI,...,EnIE) x L ~ (EIF)

L $ (EI,...,En;F);

c: Lpf(E1;E2) x L4F(E2;E3)--.---~LplP(EI;E3).

6.4.

Some canonical isomorphisms. We first consider the vector spaces L(IR;E) and E,

which are isomorphic, and we investigate whether they are also homeomorphic.

(6.4.1)

Lemma.

(I):

(a) The canonical isomorphism

L(IR;E) ~

E,

defined by #(e) = [(l), is continuous. For its inverse

map ~ ,

caraoterized by (~(x))(~) =~.x, we have

(b) 1~,r: Ee...-$L(IR;E)

is continuous;

(c) ~ : E

is continuous provided that

~L(IR;E)

scalar multiplication of E is equably continuous.

-

73

-

Proof.(a) ~ being the evaluation at l~ its continuity follows from (6.2.Z). (b) Let ~ E ~

Hence • ~ ~ = W . ~ ~E. Thus we get for any

with ~.~ ~IR:

which shows that ~ ( ~ ) ~ L ( I R ; E ) .

The assertion now follows by

(2.a.7). (c) Similarly we get, if ~ ~ E and ~.I~ ~ ]R:

and we deduce from (2.8.8) that this converges to zero on E. The rest goes as before. From (a) and (b) of the above lemma we get for an arbitrary E, using (2.9.1):

(6.4,2)

Proposition. There is a natural (*) linear homeomorphism

(IR;E)

E*

From (a) and (c) of lemma (6.4.1) we get:

(6.4.3)

Proposition. If scalar multiplication of E is equably continuous, in particular (cf.(2.9.2)) if E is equable, then we have a natural linear homeomorphism: L(~;E) ~ E.

(*) We do not discuss the categorical meaning of "natural" or "canonical", so the statement simp3y means: the isomorphism which we are considering is a homeomorphism.

-

74

-

This in particular implies that L(IR;E) is equable if E is equable. We further mention, that from (2.8.8) and (2.5.1) it follows easily, that scalar multiplication on E is equably continuous if and only if X~E implies Il.~ ~ E; this condition will be satisfied by the so-called admissible spaces considered in § 7 and later.

(6.4.4)

Proposition. If E2 is equable, then we have a natural linear homeomorphism: L(E1;L(E2;E3)) ~

L(EI,E2;E3).

Proof. (a) Let ~L(E1;L(E2;E3) ). We consider the map ~ ~--~(~), where ~(~) = b is the map from ElXE 2 into E3 defined by

(6.4.5)

b(Xl,X 2) = (~(~))(Xl,X 2) = ( t ( X l ) ) ( x 2 ) . This map b: ElXE2

p E3 is obviously b i l i n e a r . We show Further-

more, that b satisfies the conditions

(6.4.6)

& l ~E1; ~V.1B2~E2 ~

(6.4.7)

\V'1@l~E1;

~(2~E2 ~

b()[1,1~2) ~E3; b ( 1 ~ l , ~ 2 ) ~ E 3-

In fact, (6.4.6) follows easily: since ~:E 1

~LCE2;E3) is

continuous, ~1~[1 implies ~(~l)$L(E2;E3) , and hence (~(~[1))(~ 2) ~ E3. But by (6°4.5), (~(Z1))(~2)

= b(IEl; I~2).

In order to establish (6.4.7), we have to use that E 2 is equable. Therefore

~2 ~E2 implies

~2 ~- ~2 "- V . ~ 2 ~ E 2 ,

b ( l ~ l , ~ 2) = ( [ ( 1 3 1 ) ) ( l [ 2 ) _ z - ( ~ ( ~ l ) ) ( \ V ~ 2 ) =

and we obtain

(~(tV.~l))(~2).

-

75

-

Here, the filter on the right side converges to zero on E3, since t(W.1~l) ~ L(E2;E 3) by the continuity of ~ and s i n c e ~ V . ~ 2 ~ E 2. From (6.4.6) and (6.4.7) it follows now by (2.8.8) that b: ElXE2--.--~E 3 is equably continuous, and hence b~L(E1,E2;E3). We have shown so far that the map ~--~W(~)

= b has its image in

L(E1,E2;E3), i.e. we have

(6.4.8)

~: (b)

L(E1;L(E2;E3))---eL(E1,E2;E3)-

In order to show that ~ i s bijective, we now construct a map

~: L(E1,E2;E3) ---*L(EI;L(E2;E3)).

(6.4.9)

So let b ~L(EI,E2,E3). a map

We first define, for any fixed Xl~ El,

~Xl : E2.___~E 3 by

Xl(X2) b(Xl,X2). =

Since b: ElXE 2 ___~E 3 is continuous, and we have: ~xl~L(E2;E3).

~Xl : E2

We define e=~(b)

~E 3 is continuous, : E1

~L(E2;E3) by

~(x l) = ~Xl , and we thus have caracterized #(respectively by the equation

((~(b)(Xl))(x2)

~(b))

= b(xl,x2).

= ~(b) is obviously a linear map from E1 into L(E2;E3). We want to show that

~ :E1

~L(E2;E3) is continuous. So let

~l ~El" We have to show that ~(~l)~L(E2;E3) , which is equivalent to the condition \V.'I~2~E 2 ,,,,,~(~-(~i))(~ 2) ~E 3,

-

76

-

and this holds by (2.8.8), since ~(~i)(I@2) = b(~l, ~2 ) by (5.4.5). We remark that at this point it was essential that b~L(E1,E2;E3) implies according to the definition of the spaces L(E1,E2;E3) that b is equably continuous. Having shown that ~= ~(b) is linear and continuous, we know now that in fact t = ~ (b) ~ L(E1;L(E2;E3)), and that therefore ~ i s a map as stated in (6.4.9). It is easy to verify that 4(~(b)) = b for all b£L(E1,E2;E3) and ~ (~({)) =~

for all {¢L(E1;L(E2;E3)) , and

thus ~ is bijective, ~ being its inverse. (c)

We show that the map ~ of (6.4.8) is continuous. Since

is linear, we have to verify its continuity at the origin. So let Z ~L(EI;L(E2;E3)). In order to show that ~(Z)~L(E1,E2;E 3) , we have (cf.(6.1.5))to form ~ ( ~ ) ( ~ l , ~2 ) and to verify that this converges to zero on E3 provided that ~.I~i ~Ei, i = 1p2. But by (6.4.5) we have ( ~ ( ~ ) ) ( ~ 1 , ~ 2 )

= (~(~1))(~2),

and

here Z ( @ l ) ~L(E2;E3) and thus (~ (~1))(I~2) i E3. (d)

We finally show that the map ~ =~'l of (6.4.9) is continuous.

In fact, let ~L(EI,E2;E3).

to

W . ~ IEl ~

# (r) ~ L(E1;L(E2;E3)) is equivalent

(~(~))(~i)

~L(E2;E3),

and this again is equivalent to W~I~EI,

~325E2

~'~((F(T))(~I))(~

2) ~E 3,

- 77 -

and this holds since by the equation caracterizing ~ one has

((

=

I 2)"

This completes the proof of (6.4.4).

(6.4.1o)

Lemma. L~(EI~;E2) = LIF(EI~;E21r). Proof. (a) We first show that the underlying spaces are the same. If~IL'(EI';E2"), then ~: EI~_....~E2( is continuous, and thus afortiori ~: E1~ ----.~E2 is continuous, showing that t ¢ L#(EIf;E2 ).

Conversely, if ~Lf(Ell;E2) , we deduce from

the continuity of ~: Ela-.._~E 2 and (2.9.1) the continuity of [

:

(b)

E#.-

,~ E2¥.

We now show that the structures are the same. One part is

immediate: if ~ $ Lf(El~;E2~), then ~

Lf(El#;E2 ). Suppose con-

versely that~LW(El#;E2 ). Then ~ m ~ = ~/.)~$L(El*;E2). Let ~/'~l~Ele" Then )~(~i) IE 2. But since )I(151) = (W.~)(~S1) = ~V.~('S1), we have ewen:~(~l)~E2#.

This shows that]~L(E#;E2~), and since

~ ~ = \V.~ it follows that ~ IL~(E1t ;E2~).

(6.4.n)

Proposition.

If E1 and E2 are equable, then we have

a natural linear homeomorphism:

Llr(E1;t ~(E2;E3)) = Lf(EI, E2;E3).

-?8-

Proof. The map ~ of (6.4.8) being linear and continuous, we deduce by (2.g.1) that also

L"(EI;L(E2;E3))---, Lt (EI,E2;E3) is linear. The same holds for the map

~ = ~-l. Now using that

E = El@ by assumption, the result follows, since by lemma (6.4.10) we have Lf(E1;L(E2;E3)) = L~(EI;L~(E2;E3)).

(6.4.12)

Proposition.

If E1 is equable, then there are natural

linear homeomorphisms as follows: Lp(EIILq(EIIE2))~Lp+q(EIIE2); Lp$(E1;Lqg(E1;E2))~ L~+q (El;E2). The proofs are completely analogous to those of propositions (6.4.4) and (6.4.11). Instead of (2.8.8.) one has to use the analogous result which states that a multilinear map : E1 x ... x En ---~E is equably continuous if and only if R(~l'''"

@n ) ~ E provided that one of the filters @l'''''@n

converges to zero, the others being quasi-bounded. And at the place of lemma (6.4o10) one has to establish the corresponding equality

The map which we call natural is of course the following: to EQLp(E1;Lq(E1;E2))

we associate ~((~)~Lp+q(EI;E2), Q((R) being

ceracterized by (~e)(Xl,...,Xp+q)

= (~ (Xl,...,Xp))(Xp+l,...,Xp+q).

-

(6.4.13)

79

-

Proposition. The map (fl,f2) i

~l,f2]

(cf.(1.3.2))

yields natural linear homeomorphisms as follows:

CoCE;EI) x CoCE~E2) ~ CoCE~EI×E2); Lp(E;EI) x Lp(E;E2)=Lp(E;ElXE2). Proof.(a)

We first show that (fl,f2) ~ Co(E;E l) x Co(E;E 2) implies

[fl,f2] & Co(E;ElXE2). By (1.3.3) and (1.5.1) We have Ill,f2] (~) = (flxf2)(d(1~)) ~(flxf2)(~x ~) = fl(l~) x f2(B). Thus, if W.~ ~ E~ the filter on the right hand side and hence a fortiori [fl,f2] (~) is quasi-bounded, which shows that ~fl,f2] is quasi-bounded. From the equality

~[fl,f2] (a,x): (~fl(a,x), ~f2(a,x)) we deduce similarly that

A[fl,f2] (~,~) ~ ~fl(~,~) x Af2(~,~), and from this it is obvious that the equable continuity of fl and f2 implies that of Ill,f2] . We thus have shown that (fl,f2) l

(6.4.1~)

~Ffl,f2] induces a map ~:

Co(E;E1) x Co(E;E2)--~-~Co(E;ElXE2).

(b) Let, conversely, f ( Co(E;EIXE2). Then we conclude by (2.8.?) and (2.8.5) that ~iof, where TFi : ElXE 2 -.--~Ei is the projection, lies in Co(E;Ei), i = 1,2. Thus f ~('Ir~,f,

(6.4.15)

~F2.f ) yields a map

~: Co(E;EIXE2) ---~Co(E;EI) x Co(E;E2),

-80-.

and ~

is obviously the inverse of ~ . Both maps ~,~

are

linear, and so it only remains to show that they are continuous, or equivalently: (c)

continuous at the origin.

L e t ~ C o ( E ; E l) x Co(E;E2). Then ~ ~ ~l x ~2' where

~i~Co(E;Ei)"

If W . ~ E ,

further, since ( ~ r ) ( 1 5 )

we have therefore

&~(l~)

This shows that ~ C o ( E ; E l X E 2 )

~i(1~)$Ei, and

x ~ 2 ( ' 6 ) , also ( ~ ' ) ( ~ ) ~ E L x E 2.

and hence the map (6.4.14) is

continuous. (d)

Let ~Co(E;ElXE2).

We have to show that

# ~ICo(E;E1) x Co(E;E2) , or equivalently: w i ( ~ ) ~ C o ( E ; E i ) , where ~i' i = i~2, are the projection maps of the product

Co(E;E1) x Co(E;E2). Since Wi(g) =~i.g we get

which ends the proof of the First homeomorphism. The second result, concerning p-linear maps, is a corollary; one only has to remark that Ill,f21 is p-linear if and only if fI and f2 are p-linear and that Lp(E;Ei) has the structure oF subspace of Co(EX...xE;Ei).

(6.4.16)

Propositio.~. IF ~:E 1

~E 2 is linear and oontinuous,

then ~: Co(E;EI)----~Co(E;E2) defined by ~.(f) =~ -f also is linear and continuous.

-

B1

-

By (2.8.5) and (2.8.7) ~,f~ Co(E;E2). ~ is obviously linear. Let no. T~Co(E~EI) , W ~ E. Then ~ ( ~ ) ~ E I anO ~(~(15)) = (t',~')(~)~E2, hence ~o~= t . ( ~ ) ~ Co(£;E2) .

(6.4.17)

~orollary.

I f [z E1 ~

E2 is a linear homeomorphism,

then so is ~.: Co(E;EI)---~ Co(E;E2)

-

§ 5.

50

-

FUNDAMENTAL THEOREM OF CALCULUS. =

The value of a vector-valued function will be estimated by means of its derivative.

Since no norm is available, estimation

is formulated by means of convex sets, which is advantageous also in the normed case.

5.1.

Formulation and proof of the main theorem. Suppose there are two maps given: f:[~,~]

~ E and

~, E being a pseudo-topological vector space, such that the following conditions are satisfied

(51.1)

E ° separated;

(5.1.2)

f:

(5.l.3)

B a closed and convex subset of E°;

(5.14)

for almost all (*) t ~ ~ ]

[~,~@] ~

E ° and

~: ~o(,M

; ~ continuous;

, f and ~

are differen-

tiable at t and satisfy

f'(t) (5.l.5)

s ~t

..... ~ ( s ) ~

~ ~'(t).B; ~(t).

* Throughout this paper, "almost all" is never used in the sense of measure theory~ but always means: "all with at most a denumerable infinity of exceptions".

-

(5.i.6)

51

-

Theorem. Under the above hypothesis one has

f(~ ) - f(~)~ ( ~ ( ~ ) -~(~)).a. Proof.

Part i. In this part, we reduce the general case to

the following special case :

(5.i.7)

~=0

;

~ (0)

= 0 ;

f(o)

: o ;

o ¢

In order to do this, we choose a fixed point p ( B

B.

and we define

~I' ~I' ~i' ~i' Bl as follows:

ql(t) = q ( t + m )

-q(~)

~l(t) = f(t+~)

- f(~)

for t &[~l, @l] -ql(t).p

;

for t ~[~l,~l]

;

BI=B-p. One easily verifies that the validity of the conditions (5.1.1) to (5.1.5) for

e(, ~, 5o, f, B implies their validity for o(1, ~l' ~l'fl'B1"

In fact, let us check (5.1.4), the others being obvious. By the chain rule, we have: q'l(t) = w ' C & + ~ )

;

fi(t) = f ' ( t + ~ ) - q [ ( t ) . p

= f'(t+~)

Hence one concludes from f'(t) ~ ~'(t).B

fi(t)e

- W'(t+o().p.

for t G [o(,~]

~'(t+~,).B -W'(t+~,).p =M,~(t).B1

for t e

Suppose now the theorem holds for the special case.

:

[~i' l@l] "

-

Then we have f 1 ( ~ l ) ¢

52 -

~I(~I).BI .

But this, using the defini-

tions of ~ l ' ~ I ' f l ' Bl' yields immediately:

f(~) . f(01)

(~(~) -t~(e()).pE(~(~)-~(ot)).(B-p),

.

which is equivalent to the assertion of the theorem.

Having

reduced t h e g e n e r a l case t o t h e s p e c i a l

we hence-

forth

assume t h a t c o n d i t i o n s

P a r t 2.

We suppose i n t h i s

a neighborhood o f z e r o ,

(5.l.S)

f(/~)

£(4(~)

(5.1.?)

case ( 5 . 1 . 7 ) ,

hold.

pact of the proof that

and we show t h a t + ~'#

B is,

i n E°~

then one has f o r a l l

~ >0:

+~).e.

Let #'i' JP2' t3'''" be an enumeration of the points where possibly (5.1.4) does not hold (*), and let us define an auxiliary function : IR by

(5.l.g)

~(S)

=~(s)

* E.S

+ E.

Let

7n C- S

f(s) G

O b v i o u s l y one has •

0¢I.

(½)n. ~(s).B

Let ~ = sup I.

for 0 4=s ~ t ~

If 0 ~ t

z g,

there

exists t I ~ I with t At) ~ ~ ; hence f(s) G ~(s).B for 0 ~ $ ¢- t I, thus in particular for 0~s_~t,

which shows that t & I end that

f(t) •

~ (L).B. But this shows that also

fore:

I - G0,~]

~

I, and we have there-

.

* We can always take ~ and ~ among the points ~n; then we do not have to bothsr what differentiability would mean (cf.3.4).

in the endpoints ~ ,

-

53

-

We claim now that f ( ~ ) £ ~ ( ~ ) . B . I f ~= O, nothing has to be proved. If

~ > O , we use that

~

B

Since f:[~,~]-

for O , t

; E° is continuous and ~ continuous from the

left t it follows, using that B is closed in E°, that

t~ which means

(S.l.lO)

f(~ ) • Since I : [ 0 , ~ ]

~(~).B. ~ [0,~]

, we have of course

~--~.

(S.I.II)

In order to show that here equality holds, we proceed i n d i r e c t l y : we assume that

~ ~.

Then either ~÷~n for a l l n or ~ = ~m" We

show that none of these cases is possible. Case l: ~ ~ ,

V ~& ~n for all n.

This means that ~ is not an

exceptional point: f and ~ are differentiable at ~ and (5.1.4) holds for t = ~ . Thus we get f(~+h) = f(~ ) + h.f'(~) + rl(h) , ~(~+h)

=~(~)

+ h.~'(~)

f ' ( V ) ~ 4"(~).B.

+ r2(h),

where rl~

R( IR;E°);

where r 2 C

R(IR; IR);

- 54 -

Since II = I-l,1] certainly satisfies W.I I~IR we conclude: ~rl(W, [II])~E m. which means ~rl(\V,[ll]) @ ~ , ~.Tbeing the neighborhood filter of zero in E°. Since in this part 2 of the proof we assume that B ¢ ~ ,

we have also e/2.B ~ ].Fc @rl( W, [Ii]),

GI~O such that ~ r l ( I ~ ,

and hence there exists

I1) C t/2 B, and

thus (cf (3.I.i)):

l>l+'~l

for

~l(~ )G +/2. ~ .a

In the same way we conclude: there exists ~2

~2(~) ~

1,2(~,)1

~/2. A. z I

0 such that

for 1~1-~2,

-'- "12.1;~1

i.e.

for IM~'~2-

Let now ~= Sin ( ~I' ~ 2 ' ~ ' ~ )" Then we have for O ~ H

~6:

f(~+h) = f(~) + hof'(~) + rl(h) E ~(~).B + h. ~'(~).B + h. E/2.B -~(~).B Here,

the c o e f f i c i e n t s

+ ( ~ ( ~ +h)-~(~)-r2(h)).B

~(~), q(~ +h)- +(~)-r2(h)

= h.

+ h.

e/2.B.

~'(~)

and h. ~/2 are, using (5.1.5), ali non-negative. Using that for e convex set B and non-negative coefficients/u, Y ,G" one has

/~.8 +Y.B + ~'.B c(/~+~+G).B~we get therefore: f(~+h) E ( ~ ( ~ )

+~(~+h)-

= (~(~+h)

. +. ~

~(~)-r2(h)

(½)n+ +.+ _r2(h) + h ./2) .B

C (W( V +h) ++. ~.,+.>"'+ f,,,, l e

~lV~l

(IV,152)-

Observe now that (~

l~l~V 1~2) • 15 2 )0- = ((D2f(al+W~l,a24V~2)

by the admissibility at

- D2f(al,a2) ). ~2)°-~E3

assumption of E3 and the continuity of D2f

(al,a2).

Further

e ~Vl~l.(%V , ~2 ) ~ 8 r2(~J,~lXl@2) ~ -~ r2(|V,15) which proves that r2 E R(EIXE2;E3).

Hence by (3.1.4):

r2 + rlo~f 1 C R(ElXE2;E3) , and (8.2.2) shows that f is differentiable at (al,a2). The continuity assertion for Df follows from (6.3.2), because

Df(Xl,X 2) = Dlf(Xl,X 2) elTl * D2f(xl,x2 ) • 1" 2 .

-

93

-

§ g.

HIGHER DERIVATIVES.

9.1.

f" and the s~mmetry of f"(x). If u : ElXE 2

~ E 3 is bilinear and continuous, we already

know that u is differentiable at every point (4.2.3), but we are not able to prove that Du : ElXE2----~L(ElXE2;E 3) is continuous unless E1 and E 2 are equable vector spaces.

(9.l.l)

From now on we suppose that E, El, E2,... are always equable and admissible vector spaces.

(g.l.2)

Definition.

A map f: E1----~E 2 of equable and admissible

vector spaces is called twice differentiable at a point a, iff Df: E1-----~Lf(EI;E 2) exists in a E-neighborhood of a (cf. 3.4) and is differentiable at a. We write D2f(a) instead of D(Df)(a) and remark that thus D2f(a) ~ L~(E1;L~(E1;E2)).

The element which corresponds to

D2f(a) in the natural isomorphism

(6.4.2) is denoted by f"(a),

and we shall write f"(a).(s,t) instead of (f"(a))(s,t).

f"(a) is

a bilinear map : f"(a) E L2f(E1;E2).

(9.1.3)

Proposition.

If f:El-~-~E 2 is twice differentiable at a,

then f"(a), i.e. the bilinear mapping (s,t),-~f"(a).(s,t), is symmetric

(*).

n

* The same is true under the weaker condition that f: EI----~E ~_ is twice differenti~ble.

-

Proof.

Let (s,t) e ElxE 2,

g : [O,1]

of ao

-

~clR and consider the mapping

~ E2 defined by g (~)

Let I1 = [O,1] .

94

= f(a+~.s+

At) - f(a+ ~ . s ) .

By assumption Df exists in a El-neighborhood U

Hence a+\V(Il.S+t) ~ U, because

~V [Il.S+t ] ~ ~Vs+ ~Vt~ E1 .

In other words, there exists @l > 0 such that a + ~.s

+ ~.t £ U for

I ~ ~ ~l and

g is differentiable in [O,l 3 for each

~ ~ [O,q

, and therefore

I~I _z-i1.

By (3.3.1) and (4.3.2), we get g'(~)

= CDf(a+~s+

~t) - Dr(a+ ~ s ) ~ . ~ s .

o Inside any neighborhood of zero in E 2 we can choose a neighborhood W which is absolutely convex and closed; this means that -

=W.

-W=W=

We show that there is ~2 ~ 0 with

(9.1.4)

(Df(a+ ~ s +

e2 ~ ~i such that for

~t) - Of(a) - O 2 f ( a ) . ( ~ s +

and ( D f ( a + ~ s )

~t)).s

- Df(a) - D 2 f ( a ) . ( ~ s ) ) . s

I~I ~ e2"

( &W

e &W.

From the differentiability of Df at a we conclude also that eRDf(a)

. (\V, [Ii.s+t ~ )~ L(E1;E 2) hence, because

~RDf(a)

. (W,

[Il.S + & ] )

• s _~W]which yields (9.1.4).

Now we get by multiplying with ~ g'(~).l

~VS~El~

and subtracting

- ((D2f(a). ~t) o u Xs).l e 2 ~2.W where u &s : IR ----?E2

is defined by u ~s (~)

= ~ . ~s.

-

95

-

By (5.3.1) we get g ( 1 ) - g(O) - ( D 2 f ( a ) . g(l)

~3

~t).~

s ¢ 2A2.W .

- g(O) b e i n g symmetric i n s~t~ we can f i n d

~ ~2 ~ such that for each I~I ~

g(l) - g(0) -

s).

t

2

e3 > 0 w i t h

~3 : 2W;

hence by subtraction (D2f(a). ~t) . ~ s - (D2f(a). ~s). ~t ~ 4 ~2W for each

I~I ~-~3" Finally

(D2f(a).t).s

- (D2f(a).s).t

: f"(a).(t,s)

- f"(a)o(S,t)

~ 4 W.

This proves the symmetry, since E02 is separated. 9.2.

f(P) for p ~ 1. By induction on p we define p-times differentiable mappings

as follows : A map f: E1 ~

E2 of equable and admissible vector

spaces is called p-times differentiable at a, iff f(p-l): E1 _ _ ~ L~p-l(E1;E2) exists in a El-neighborhood of a at a.

and is differentiable

Then by definition D(f(P-l~(a) ~ L~(E1;Lp~I(E1;E2) ). By (6.4.12)

this vector space is linearly homeomorphic to L~(E1;E2). We write f(P)(a) for the p-linear mapping thus corresponding to Df (p-1)(a).

(9.2.l)

proposition.

If f : E1 _-~ E2 is p-times differentiable

at a point a, then for any fixed (s2,s3,...,Sp) , the deri~t~ e_.._~f (p_l) vative~JBf the mapping x (x).(s2,s3,...,ep) of E1 into E 2 is the linear map sI

~f(P}(a).(Sl,S2,...,Sp).

-

Proof. Let UCLn(EI;E2).

96

-

Then the mapping u ~

?u(tl,t2,...,tn) ,

where (tl,...,tn) is fixed, is linear and continuous by (6.2.2). The result follows from (3.3.1), (4.2.1) and the equality

(Df(P-1)(a).sl) . (s 2, .. . ,Sp) (9.2.2)

Proposition.

=

If f: E1

f(P)(a)

. (Sl,S2,...,Sp) (see(6.4.12)).

~E 2 is p-times differentiable at

a ~ El, then the p-linear mapping f(P)(a) is symmetric. This is a consequence of the two preceding propositions. The induction proof can be found e.g. in [3] p.177.

(9.2.3)

Proposition. If f: E l S E

2 is p-times differentiable and

f(P): E1----~L~(E1;E2) is q-times differentiable, then P f: E I ~ E Proof.

f(P): E1

2 is (p+q)-times differentiable. ~ L~(E1;E2) being q-times differentiable,

(f(p))(q-1): E1 _.._~Lq~_I(E1;L~(E1;E2)) by (6.412) also f(p+q-1): El_. ~

is differentiable. Hence

L p+q-1 ~ (El;E2) is differentiabie,

which ends the proof. We call a mapping f : E 1

~E 2 infinitely differentiable~

~ff it is p-times differentiable for all p ~ . (9.2.4)

Example. If u : ElXE 2

~E 3 is bilinear and continuousp

then u is infinitely differentiable. We already know (see 4.2.3) that Du exists throughout ElXE 2 and is linear.

- 97 -

To show that Du : ElXE 2 ~ L $ ( E l X E 2 ; E 3 ) is continuous, let ~l x ~2 ~ E3xE2' \V(J~lX~2 ) ~ ElXE 2. Then

Ou( ~j., ~2 )

.

(,,a.I, ~2) :

by (g.l.l))

(2.g.2),

u(~(z,,,~2) +

u(,A.z, ~.2) ~, E3

(2.8.8) and (2.3.7).

Hence by (4.2.1) Du : ElXE 2 ~ L m ( E l X E 2 ; E 3 )

is diffsrentiable, and

(Du)'(Xl,X2) = Du for each (Xl,X2) E EIXE 2.

Therefore

u" : ElXE 2 -----~L2~(ElXE2;E3) is a constant mapping and thus differentiable by (3.2.3). The same is true for u (k), k ~ 2 , which ends the proof. (g.2.s)

Proposition. If fl: E ----~El and f2: E ----~E2 are p-times differentiable at a, then the mapping Ill,f2] : E

7EIxE 2

is p-times differentiable at a, and [fl,f2](P)(a) = [fl(P)(a), f2(P)(a)l. Proof.

For p = 1 this is (4.4.5).

Suppose (g.2.5) for p and let

fl,f2 be (p+l)-times differentiabls at a.

We have to prove that

x ~-'-~[fl,f2~(P)(x) is differentiable at a. By the induction hypothesis we have [fl,f2] (P)(x) = fl (P)(x), f2(P)(x)], which is an element of L p (E;ElXE2). This space is linearly homeomorphic to Lp(E;E1)XLp(E;E2) by (6.4.13), the corresponding element being the pair (fl (p)(x)' f2 (p) (x)).

-

98

-

It is therefore sufficient to prove that x ~----~(fl(P)(x), f2(P)(x)) is differentiable, which is now a consequence of (4.4.5), because (fl(P)(x), f2(P)(x)) = ~fl (p), f2 (p)] (x) by (1.3.2), and ~fl (p), f2(P)j '(x) = [(fl(P))'(x),

(f2(P))'(x)] corresponds to

[fl(P+l)(x), f2(P+l)(x)] .

(9.2.6)

Theorem. If f : E1

~ E2 is p-times differentiable at a~

and g : E2---~E 3 is p-times differentiable at b = f(a), then g6f : EI _.--~E3 is p-times differentiable at a. Proof. For p = 1 this is the chain rule (3.3.1). Suppose (9.2.6) proved for p, and let f and g be (p+l)-times differentiable. By (9.2.3), gof is (p+l)-times differentiable, if we show that (gof)' is p-times differentiable. write (gof)' = c o ~f',g'°f]

By (1.3.2) and (3.3.1) we can

, where the bilinear map

c: L~(E1;E 2) x L~(E2;E3) ----~L~(E1;E3 ) is infinitely differentiable by (6.3.3), (gol.1) and (9.2.4).

The mapping g'of is

p-times differentiable by the induction hypothesis. Further by (g.2.5), the same is true for [f',g'.f] • Finally, applying the induction hypothesis completes the proof.

to the maps [f',g'of] and c, one

-

§ i0.

99

-

Ck-mAPPlNGS.

In § 2 the vector space Co(El;E2) was introduced, and in § 6 endowed

with a pseudo-topology. We recall the definition:

f~Co(E1;E 2) iff : W~,@~E

l :~f(~),

$ co(E l

Z 3 f ( ~ , X ) ~ E2;

iff :

~ V ~ E l . . ~ ~(~:L)~ E 2, We now introduce, always assuming that El, E2 are equable and admissible (cf. (9.l.l)), a class of mappings, called Ck-mappings; in the case of finite dimensional spaces, these are exactly the k-times continuously differentiable mappings.

i0.io

The vecLo r space Ck(EI;E2).

(lO.l.l)

Definition: For any k ~ o

(.) we call f: E1-----~E2 a Ck-map

and write f~Ck(E1;E2) , iff f is k-times differentiable in El and f(k)e Co(E1;L~(EI;E2)).

(io.i.2)

Propqsition. If E1 is finite dimensional and E2 normabla, then f~Ck(EI;E2)

if and only if f is k-times continuously

differentiable. This follows by (2.8.6), (2.6.1), (2.6.2), (2.5.2) and (2.8.3).

:

0,I,2,...

; for k : O, we define f(o) : f and

Lo~(EI;E 2) = E2# f o r k = i ,

f(1) = f, and LIQ(EI;E2 ) = L@(E1;E2).

-

(I0.I.3)

I00

-

Proposition. If f(Ck(EI;E2) , then f(n) ~ Co(EI;~(EIIE2) ) for n = O,...,k. Proof. The case k = 0 is trivial. k = I.

We prove in detail the case

We then know that f'e Co(EI;L$(E1;E2 )) and we have to show

5o l e t \VJ~E 1•

that fG Co(El;E2). and ~= [ O , l ] . ~

.

Let us put fo(X) = f(x) - f(o)

Since f ' = f ' is quasi-bounded and~.~ = W . ~ E I , o

~.f~(~)~L¥(EI;E2).

Hence, using (2.6.3)

(\V.f~(1~))(~)

= (W.fo)'(~).~E

2.

Since by (1.5.2) and (5.3.6) we have,

~.f(~)

~ W . f o ( ~ ) + w.f(o) ~ (W.fo)(~) + W.f(o)

~((Wfo),(~).~) °-

+ w.f(o)

we conclude, E2 being admissible, that W . f ( ~ ) ~ E2.

This shows

that f is quasi-bounded; i t remains to show that i t is equably continuous. Since

So let

W A, ~ ~E I.

~f(a,x) = f'(a).x + Rf(a).x,(1.5.2) yields ~f(~,~)

~ f,(~).~

+ Rf(~).~

.

Here, f'(w~).~ ~ E2 by (6.2.2), (g.l.l), (2.g.2) and (2.8.B), since

~.f'(~) ~ L(E1;E2).

The second term is estimated by means

of (5.3.6), which can be applied to the filter Rf(~) = ~ s i n c e (Rf(s))(o) = 0 for all a ( E1 .

We thus obtain, using that

(Rf(a))'(x) = ~f'(a,x) and that ~ ~ ~ = W ~

= [O,1] . ~

~ E1

-

i01

-

by the equability of E1 :

(Rf(~))(~) ~ (Rf(~)).~ ~(~f,(~,~)(~))o-. The equable continuity of f' and the admissibility of E2 implie that the right hand side converges to zero on E2.

This completes

the proof for k = i. If k > l~ we deduce from f(k)( Co(El,.LW[ k~EI;E2) ) by means of the linear homeomorphisme (6.4.12) and (6.4.17) that (f(k-1)),~Co(El;L,(El;Lk_~(E 1;E2))), and the same arguments used before yield f(k-l)~ Co(El;Lk~l(El;E2))"

Repeating this

k-times one completes the proof (see also (7.4.6) and (7o2.3)).

(lO.l.4)

.Cp,rollary, C_k+n(EI;E2)¢ C.Ck(EI;E2), For k > i, the proof is the same in view of the linear homao-

morphism Co(EI;L~(EI;E2)) ~w Co(EI;L'(E 1 ;Lk~l (El;E2))) where (f(k-l)), corresponds to f(k) and Lk~_I(EI;E2 ) is an admissible vector space.

For details see (6.4.12), (6.4.17), (7.4.6) and

(7.2.3).

10.2.

(lO.2.l)

The structure of Definition.

Ck(EllE2),

~Ck(El;E2)

iff :

[(n) ~ Co(EI~,,(EI~E2)) for n -- 0,I,2,.. ,~k.

-

102

-

We remark that by (I0.1.3) ~(n) is in fact a filter on the indicated space. Lemma. The inclusion (cf. (10.1.4))

(10.2.2)

Ck+n(E1;E 2) c Ck(E1;E 2) is continuous. This follows at once by (10.2.1).

(lo.2.3)

Lemma. The mapping f ~_.~f(n) of Ck+n(E1;E2) into Ck(E1;Ln~(E1;E2 ) is linear and continuous for each n ~ ~o. Proof. By (6.4.12) and (6.4.17), the corresponding element of f(n+k) E Co.(EI;Ln~k~(EI;E2 )) is (f(n))(k)~Co(E1;L~(E1;L~(E1;E2)))" hence f(n) ~ Ck(EI;Ln~(E1;E2)).

Let now ~'~CK+n(E~;E2).~

Then

by definition ~(P)~Co(E1;Lp@(E1;E2) ) for p = 0,1,...,k+n. By (6.4.17) and (6.4.12) we have

~o~ ~ ' ~ ~ p = n,n+l,...,n+k.

~Co(~~ o ~ ~" ~ ~ ~) ~o~

Hence (~(n))(P-n)~Co(E1;Lp~n(E1;Lg(E1;E2) ~

for p = n,n+l,...,n+k, which shows that~(n)~Ck(E1;Ln@(E1;E2)). This proves the continuity.

(lO.2.4)

Proposition.Ck(E1;E 2) is an admissible vector space. Proof. We have to veri~y that Ck(E1;E 2) satisfies the compatibility conditions (2.4.2) and the admissibility conditions (cf.(7.1.1)). By (9.1.1), (7.4.6) and (7.2.3), Co(E1;Ln~(EI;E2 )) is admissible for n

= O,...,k°

-

103

-

Observing that the mapping f ~----)f(n) is linear, one easily verifies the compatibility conditions for Ck(E1;E2).

We next

verify the three admissibility conditions. o a) The inclusion i : Ck(El;E2) 9 nuous by (10.2.~) and (2.9.1). Let f ~

-Co°(El;E2) is contiCk(E1;E2), f ~ O.

Then f E Co(El;E2) , and by (7.4.6) there is a convex neighborhood V of

rl

o 6 Co(E1;E 2) with f ~

V.

Hence by continuity

i-l(v).

i'l(v) is a convex neighborhood of o ~ Ck(E1;E2), and f Hence Ck(E1;E 2) is separated• b) Let ~ C k ( E 1 ; E 2 ) .

By (lO.2•l) and (7.4.6) we have

~---~) ~ Co(E1;Ln~(E1;E2)), n = O,...,k. map f ; ~(n)

~f(n) ((10.2.3)) implies that (~-)(n)z-~-~), because

C -F~ )

for each F ~qJ7 (see also (2.9•)~). Hence ~ C

c) Let ~ C k ( E 1 ; E 2 ) . f ~

(10.2.s)

But continuity of the

~ f(n)

k(E l~E2).

By the linearity of the mapping

it follows that (~) (n) = ~ )

Hence ~ C k (E1 ;E2 )

Remark• It seems that in general Ck(E1;E2) is not equable, not even if E1,E 2 are topological.

For this reason we later on

consider Ck(1;E2), ~E if the equability condition is required.

-

i04

-

Combining (6.1.8) and (10.1.2) we obtain the following generalization of

(6.1.g)

(10.2.6)

Proposition. If E1 is a finite dimensional vector space with its natural topology and E2 a normed vector space, then Ck(E1;E2) is the space of k-times continuously differentiable maps from E1 into E2 with the topology of uniform convergence on bounded sets of the derivatives of order O~l,...,k.

103

c ~ (ELSE2) The admissible vector spaces Ck(E1;E2) furnish a projective

system of pseudo-topological vector spaces (cf. (2.3.5)), where [I, ~ =

I~°,~.~

We define C ~

and the inclusion maps are continuous by (10.1.4).

(EI;E 2) to be the projective limit of this system.

Hence by (2.3.5) and (7.3.2)

(lO.3.1)

C~(E l~F2) : ~

C#E l~F2)

~" ~ C ~ ( E 1;E 2)

iff

k=o

(lO.3.2)

~ Ck(E1;E 2) for each k E ~ ° .

(lO.3.3)

Remark. It is readily verified that the propositions (lO.1.2), (10.1.3), (10.2.2) and (10.2.3) are also true for k = ~ .

-

(lO.3.4)

...proposition. .

105

-

If E1 is a finite dimensional vector space

with its natural topology and E2 a normed vector space, then C=~(EI;E2) is the topological projective limit of the topological vector spaces Ck(EI;E2). Proof.

In view of (I0.2.6), it is sufficient to show that the

pseudo-topological projective Iimit of any projective system of topologicaI vector spaces Ei, i £ I, is the same as the topological projective Iimit, which means that the coarsest pseudo-topology on E =

(~ iEI

Ei for which the incIusions fi : E

nuous, i s a topology ( * ) . we have

~ E i are conti-

By (2.3.3) and since @~E ~

fi(t[)~Ei,

:

ftCsup

= sup

Hence

sup

10.4.

Higher order chain rule.

~£,

(X))

= sup

which by (2.4o4) proves the assertion.

In order to prove the chain rule for Ck-mappings ~i0.4.7~,

we need some auxiliary results.

(lO.4.l)

Proposition. Let f: E1 ----~E 2 be C k.

Then :

f is Ck+ p ~_..~f(k) is Cp.

(*)

For inductive limits, the situation is different: for topological vector spaces, one has to distinguish between the topological inductive limit and the pseudo-topological inductive limit, the latter not necessarily being topological.

-

Proof. Let f ~ C k+p(E1,"E2).

106

-

Then f is (k+p)-times differentiable

and from f(k+p)~ Co(E1;Lkf+p(El;E2) ) we deduce by the linear homeomorphisms (6.4.12) and (6.4.17), that

(f(k)) (p) ~ Co(E1 ;Lp~(E 1 ;Lk~ (E1 ;E2) )) and that

f(k) : E1 ___.~Lk~(E1;E2) is p-times differentiable.

Conversely, if f(k) is Cp, then f is Ck+ p for the same reasons and by the assumption.

(lO.4.2)

proposition.

Let ~

be a filter on Ck+p+l(El;E2). Then:

~ Ck+p+l(E1;E 2) ~----~[~'~Cp(E1;E2)(*)

and

%

proof. Let • ~ Ck+p+l(E1;E2).

Then • $ Cp(E1;E2) by (10.2.2) and

~(p+l) ~Ck(E1;Lp+l~(E1;E2) ) by (10.2.3).

Conversely, ~

Cp(£1;£2)

means that ~(m) ~ Co(E1;L:(EI;E2)) for m = O,l,...,p and ~(p+l) ~ Ck(E1;Lp+14r(E1;E2)) implies ~(p+l+n)~ Co(E1; Lp .~,~(E.;E2)) ?~?ll

&

for n = O,1,...,k by (6.4.12) and (6.4.17), which ends the proof.

(10.4.3)

Propositio n. If ~ ~ L(E1;E2) , then ~, : Ck(E;E1) ---~Ck(E;E2) defined by Proof.

~.(f) = [of is linear and continuous.

For k = 0 this is (6.4.16).

k and let f ~ Ck+l(E;E1).

Suppose the assertion for

First we show that ~of ~ Ck+l(E;E2) .

By (3.3.1) we have (~of)'(x)

= ~of'(x).

The linear mapping

~. : L~(E;E1 ) ---~ L#(E;E 2) is continuous by (6.3°3)°

(*)

To be exact, one should write: i(~) ~ Cp(E1;E2) , where i : Ck+p+l(E1;E2)

~ Cp(E1;E2) is the inclusion.

-

i07

-

Further f'E Ck(E;L~(E;EI)) by (i0.2.3). Applying the induction hypothesis to ~. and f', we get (~of)' = ~.o f'~ Ck(E;L~(E;E2)), hence ( ( t . f ) ' ) ( k ) ~ Co(E;Lk~CE;L~(E;E2))~ and (eof)(k+l)~ Co(E;Lk~I(E;E2)) by the linear homeomorphisms (6.4.12) and (6.4.17). Let ~Ck+l(E;E1)o

Then ~.~is a f i l t e r on Ck+I(E;E2) by the prece-

ding result ~o~Co(E;E2) by (6.4.16) and (10.2.2)~ (Eo~)'

=

~. o ~ ' ~ Ck(E;L$(E;E2)), because ~'~Ck(E;L~(E;E2)) by

(10.2.3) and g :

~.og = (~.).(g) of Ck(E;L$(E;E1)) into

Ck(E;L~(E;E2) ) is continuous by the induction hypothesis. The induction proof is completed by (10.4.2).

C10.4.4)

Corollary. I f ~: El

~ E2 is a linear homeomorphism, then

so is e. " Ck(E;EI)--'-'~Ck(E;E2)"

(lo.4.s)

Proposition. The mapping ~: Ck(E;E1) x Ck(E;E2)-----~Ck(E;ElXE2) defined by ~(fl,f2) = [fl,f2~

(see (1.3.1)) is a linear homeomorphism.

Proof. [fl,f2~ is k-times differentiable by (g.2.5). By definition (lO.l.1): (f~k) f~k))~ Co(E;Lk~(E;EI) ) x Co(E;Lk$(E;E2)). By (6.4.13) thi~ space is linearly homeomorphic to Co(E;L~(E;E l) x L~(E;E2)), hence also to Co(E;Lke(E;EIXE2 )) by (10.4.3), the corresponding

element being Ell'f2](k).

-

10B

-

In order to prove the continuity of K , morphisms and (6.4.14).

one uses the same homeo-

The continuity of ~-l

follows from

(10.4.3), since ~ - l ( f ) = (~TlOf, ll-20f).

(i0.4.6)

Proposition. continuous,

If u : EIXE 2

~ E 3 is bilinear and

then u is Co..

Proof. By (2.8.10) and (g.l.1), u is Co . nitely differentiable,

by (2.8.7).

By (9.2.4), u is infi-

u' is linear and continuous, hence C o

u (k) is a constant map for k ~

2, hence obviously

Co .

(i0.4.7)

Theorem. If f ~ Ck(EI;E2) and g (Ck(E2;E3) , then gof ~ Ck(E1;E3). Proof. For k = 0 this is (2.8.5). let f and g be Ck+ 1. tiable.

(lO.4.B)

Suppose the theorem for k and

By (9.2.6), gof is (k~l)-times differen-

We assert that (gof)' ~ Ck(EI;L~(EI;E3)).

By (3.3.1) we get (gof), = c off' , g'6f], where c :

L$(EI;E2) x L¢(E2;E3)~L#(E1;E3) is C~. by (6.3.3) and (I0.4.6). g' ~ Ck(E2;L~(E2;E3)) and f' ~ Ck(EI;L~(EI;E2)) by (10.2.3).

f ( Ck(EI;E2) by (10.1.4).

Applying the induction hypothesis

to f and g', we get g'of ~ Ck(EI;L$(E2IE3) ).

Hence

~f', g'of] ~ Ck(E1;LC(E1;E2 ) x L~(E2;E3) ) by (10.4.5).

-

log

-

Applying once again the induction hypothesis to c and [f',g,ef] ,

we

get (I0.4.8).

From this we conclude

((gof),)(k) ¢ Co(E1;Lk#(EI;L$(EI;E2))~, hence

(g.?)(k+l)

Co(EI;Lk+I~(EI;E2)) by the linear homeomorphisms

(6.4.12) and (6.4.17).

One could establish a formula expressing

(g.f)(k) by means of the derivatives of f and g; the formula is the same as in the classical theory.

-

§ II.

llO

-

THE COMPOSITION OF Ck-mAPPINGS.

It will be shown that the composition map c : CJ(E1;E 2) x C~(E2;E 3) ......... > C~.~(E1;E3) is not only continuous (*), but even C ~ o We thus get a non trivial example of a

C_

_mapping between spaces which in general are infinite

dimensional and not topological.

Since the notion of a Ck-mapping

was only defined for maps between equable spaces, we have to consider the spaces C~(Ei,Ej)

(cf.(i0.2.5) and (2.6.4)).

They coincide with the spaces C ~ (Ei;E j) if e.g. the spaces E1,E2,E 3 are finite dimensional (cf.(lO.3.4)).

II.I.

The continuity of the composition map.

(ll.l.l)

proposition.

Let uGL(EI,E2;E3).

mapping ~ : Ck(E;EI) x Ck(E;E 2)

Then the bilinear ~ Ck(E;E3) defined

by ~(f,g) = u .~,g] is equably continuous. Proof.

By (10.4.5), (10.4.6) and (10.4.7), u~(f,g) e Ck(E;E3).

To prove the continuity of "~, u we show

(1) ~Ck(E~EI), W~ ~Ck(E~E2) ~

~(~,~) ~Ck(E~E3).

(2) W~Ck(E~E1), ~ C (E~E)

0(~,

(*)

Ck(E;E3).

This continuity statement is not eas~y comparable with the continuity result of (ll.l.~); however the latter is used in order to obtain the differentiability of c.

-

III

-

which will end the proof by (2.8.8). Let k = O, IVJ~E. Then by ( 1 . 5 . 2 ) and ( 2 . 8 . 8 ) we have in both cases (~(~,~))(J~) -~ u (~'(~), ~ (J~))~E 3 because either

~(~)~EI,\V~(~)~E 2 or W~(~)~EI,~(~)~E 2 Suppose (ll.l.l) for k and let us consider the first case:

~Ck+I(E~E1), Then by ( 1 0 . 2 . 2 )

~v ~Ck+I(E~E2)

and the p r o o f f o r k = 0 i t

0(~, ~)~Co(E;E3). (ll.1.2)

follows

that

We assert that

@ ( ~, ~ ))' ~ Ck(E;L~(E;E3)) By (6.4.11) and (g.l.l) we have the linear homeomorphisms

(ll.l.3)

L$(E1,E2;E3) ~ L$(EIIL~(E2;E3))" L$(E1,E2;E3) ~

L$(E2;L~(E1;E3)).

We denote by uI resp. u2 the linear mappings thus corresponding to u.

~

This means U(Xl,X2) = Ul(Xl).X 2 = u2(x2).x 1

o~oo~

u~

~o ~

@(~ ~))

o ~.hu~

+ 0.~

u~ ~]

where b i s the c o m p o s i t i o n map o f l i n e a r

and c o n t i n u o u s mappings,

i.e.

which i s b i l i n e a r

b(u,v)

= you, d i s c u s s e d i n

(6.3.3),

equably continuous. By (1.5.2) the above equality yields

( ~ ( ~ , ~ ) ) , ~_~(~,,UlO~) + ~(~ ,u2. ~) where IVy' = (IVy)' ~Ck(E;L~(EIE2)) by (10.2.3)

and

-

112

and UlOqJr= (Ul).(~-)$Ck(E;L~(E2;E3))by (i0.2o2),(ii.I.3) and (10.4.3). For the same reasons ~' ~Ck(EIL~(E;EI)) and yd(u24~) = u2°\V~Ck(E;L~(EI;E3)). By (6.3.~,(2.8.8) and the induction hypothesis, applied to the bilinear mapping b, it follows that b(~lJUl°~) ~Ck(E;L~(E;E3 )), and ~(~',u2o ~) ~Ck(E;L~(E;E 3)). Hence the above inequality yields (10.4.9). Applying (10o4.2) for p s 0 this proves that ~ ( ~ , ~ ) ~Ck+l(E;E3). If W~Ck+I(E;E1),~Ck+I(E;E2),

then exactly the same

proof shows that "U(~-,q) ~Ck.I(E;E3). (11.l.4)

Theorem. The mapping c: Ck(E1;E2) x Ck(E2;E3)--~Ck(E1;E 3) defined by c(f,g) = gof is continuous. Proof. We use induction on k. Observe first that

(11.1.5)

÷

(l)

k

=

o.

Let ~ x ~ C o ( E I ~ E

2) x Co(E2;E3),Wd~E1. By (1.5.2),

the above equality yields

By assumption g is Co, hence ~g(fCd~),~'(~z[)) ~E3, because ~Vf(~)~E 2

end

~ ( ~ ) ~ E 2. Further ~V(f(d~) + ~7(d~)) -~-

\Vf(J~) + ~V~(~) by (1.5.2), hence also ~ ( f ( ~ ) + ~ ( ~ ) ) ~ E 3 , which ends the proof for k = O.

-

(2)

113

-

Suppose the theorem for k, snd let

x Ck÷l(El,E2) x

Ck+I(E2;E3).

We assert that

(n.l.6)

(z~c((f,g),(~ ,~ ))),j,Ck(EI,d(EI~E3)). We again denote by b : L~(E1;E2) x L$(E2;E3) ~ the composition map discussed in (6.3.3).

L¢(E1;E 3)

Using (3.3.1) and

(ll.l.1) one gets

(n.1.7) where ~ is the map associated to b according to (ll.l.1) and c in the right hand side expression is the composition map: Ck+l(El;E2) x Ck(E2;L~(E2;E3))----gCk(E1;L~(E2;E3)). Applying the induction hypothesis to this map and using (i0.2.2)

and (I0.2.3), one gets

(n.1.8)

Ac((f,g'),(~, ~')) ~ Ck(EI;L' (E2;E 3))o ~Vf~Ck+l(E1;E 2) by the compatibility condition (4) of (2.4.2), hence (Wf)' = ~Vf'$Ck(EI;L*(EI;E2)) by (10.2.3). ~ being equably continuous by ( l l . l . 1 ) , this proves that

(n.l.9) Using the equality ( g ' + ~ ' ) , ( f + ~ ) = A c ( ( f , g ' ) , ( ~ , ~ f ' ) ) + g'of and (1.5.2) one gets

~v.(g,+ ~,)o(f+~) ~

-

114

-

by the compatibility conditions of (2.4.2), by (10.4.7) and

(ll.l.B). ~' ~ Ck(EI;L~(EIIE2)) by (10.2.3), hence also

(11.1.1o) The equality (ii.I.7) yields

by (11.1.9) and (ii.i.10), which proves the assertion (ii.I.6). The proof for k = 0 also shows that

~ c((f,g),(~,(] ))$ Co(E~E3), hence applying (10.4.2) for p -- 0, one completes the induction proof.

11.2.

The differentiability of th e composition map. In order to prove the general theorem, we establish some

auxiliary results.

(ii.2.1)

Proposition. Let g~Ck+l(E2;E3). g.: Ck(E1;E 2)

Then the mapping

~Ck(E1;E 3) defined by g.(f) = gof

is differentimble throughout Ck(E1;E2) , and

(g.)'(f).~

: ~(g'of,~).

-

Proof.

115

-

Consider the expression rg(~) = g.(f+~) - g . f -

where e : L~(E2;E 3) x E2 ~ in (6.2.3).

song'.f, ~ ,

E3 is the evaluation map discussed

According to (ll.l.1) we use the notation eo [g',f,~] = ~(g'of,~).

By (ll.l.1) the linear mapping ~ ~ ( g ' o f , ~ )

is continuous,

g,of being an element of Ck(EI;L~(E2;E3)) by (10.4.7) and (10.2.3). The proof is complete if we show that

(ll.2.2)

N

~ R(Ck(EI;E2); CM(EI;E3))"

We use induction on k. (A) Let k = O, ~VJ%~Co(E1;E2) and \VIS~E 1.

Then by (5.3.5),

(g.l.l), (1.5.2) and using the equality

(11.2.3)

@rg g, (f(x)).

Hp

because (Rg(f(x)))'(h) = A g,(f(x),h) and (Rg(f(x))~(O) = 0 for each x,h 6 E1 • This p r o v e s that Org(~V,~) ~Co(EI;E3). (B) Suppose (ll.2.1) for k and let g ~ Ck+2(E2;E3), \VJ~Ck+I(E1;E2).

-

116

-

We a s s e r t :

(ll.2.~)

(Org(~/, J~)) ' ~ Ck(EI;L~(EI ;E3)), which will end the proof by (A) end (10.4.2)o Using (11.2.3) and the chain rule one finds that the derivative of the mapping x ~ e r g ( ~ , linear map

•

~). x at a point x E E 1 is the

-~ ((~,.~~,,~)~x~ -

(o,.~)~x~ ~,~x~.~

- ~',(~x)). (~, ~).~, 4 ~xO. By

(g.l.3) one g e t s :

~,,(~x)).(~,(x).o,4 ~x)) =

~(D2g of, ~)(X)of' (x)~ .s, hence

(e,g(r).( ~, 4))' (x) =

(g,o(f+~) - g,of. ~(o2g,r,~))(x)of,(x)

+ (g'-(f+~) -g'-f)(x)o~'(x),

which yields

where (g').(f) = g'of end (11.2.5)

rg,(V) = Q'o(f+v) - g'of - ~((g')'of,~). By (I.5.2) we obtain therefore

(ll.2.6) Since g' ~Ck+ I(E2;L~(E2;E3)), the induction hypothesis applied to (g'), yields : erg,(W,v~)~ Ck(EI;L~(E2;E3))I further by (3.2.5) and (2.6.3):

-

117

-

Applying (Ii.I°i) to the bilinear map b and using (2.8.8) one proves (11.2.4) by the inequality (11.2.5).

(11.2.7)

Proposition. The mapping c: Ck(E1;E 2) x Ck+I(E2;E3)-----~ Ck(E~;E3) is differentiable throughout Ck(E1;E2) and c'(f,g).(~,~ ) : ~(g'o?,~) +~of Proof.

We verify the assumptions of (8.2.1).DlC(f,g).~ = ~(g'°f,~)

by (11.2.1). D2c(f,g ).~ = ~of, because c is linear in the second variqble and continuous by (ll.l.4). Ck(E1;E 3) is an admissible vector space by (10.2.4). DlC : Ck(E1;E2) x Ck+l(E2iE3) tinuous, because W~Ck(E1;E2)

The mapping

,L(Ck(E1;E2);Ck(E1;E3)) , is conand ~'x~ ~Ck(E1;E2) x Ck+l(E2;E3)

imply (DlC(f+~-,g+~) - Dlc(f,g))(~ ) = ~ c ( ( f , g ' ) , ~ x

~'),~

Ck(E1;E3) by (ll.l.1) and (2.8.8). (11.2.8)

Lemma.

If E2 is an admissible but not necessarily

equable vector space, f : EI___.-~E 2 is differentiable and f' :El~-----*L(EI~ ;E2~ ) is continuous, then f: Ele Proof.

~E2* is differentiable.

Let a ¢ E1 . By assumption f(a+x) = f(a) + ~(x) + r(x) where

¢ L(E1;E 2) and r E R(Ei;E2). Since by (2.g.1) we have L(EI;E 2) c L(Ela;E2~), it only remains to show that r~R(El~;E2~).

-

118

-

So let ~V15~ EI~. Since r (0) = 0 and r'(x) = Af'(a,x), we deduce °-

convo oo

zero on E2~ , because by the continuity of f' we have 4f'(a, ~V ~) ~L(El~;E2 m) and because E2~is admissible (cf.(7.2.3)).

(ll.2.g)

Lemma.

If ~VJ~Ck(EI;E 2) and ~Ck(E2;E3),

then

° ~ i Ck(EI~E3)" Proof. Let k = 0, WI~ ~E 1. Then (~oJ[)(~) = ~ ( ~ ( ~ ) ) ~ E 3 , because \V(J%(~)) = (~VJ%)(I~)~ E2. let ~V~Ck+I(EI;E2),

~ ~Ck+I(E2;E3).

Suppose the lemma for k and We have (g6f)' = ~(f',g'of),

hence by (1.5.2) (~oJ~)' _~ b (J[', ~'o~) which converges to zero on Ck(E1;L$(E1;E3) ) by (10.2.3), (ll.l.1), (2.B.8) and the induction hypothesis.

The proof for k = 0 also shows that ~oJ[~Co(E1;E3),

hence the result by (10.4.2).

Since ~ equable implies ~oJ[ equable,

we have :

(il.2.10)

Corollary.

If \V~CMN(E1;E2) and ~ ~ Ck~(E2;E3), then

~o~cM~(El~E3). (ll.2.1Z)

Lemma.

If ~Ck+I(E2;E3)

and ~V~l, IVJ~2~Ck(E1;E2 )'

then @R~.(Jll).( ~],~2 ) ~ Ck(E1;E3). Proof.

The equality Zl(g').(f, t~) = g'o(f+~) - g'of yields

~ ( ~')*(J~l' ~V~2) -~- ~'°(J~l+\V~t2) " ~'~ J~l which converges to zero on Ck(E1;L~(E2;E3)) by the preceding lemma and (10.2.3).

-

From

119

the equality ( c f . ( l l . 2 . 1 ) )

one deduces:

-

d(g.)'(f, ql )

A(~,)'(dtl,~VA2).~ 2

:

"~2

= ~L~(E4XEs;ET) and by

~ ; : Lf(Es;E7)

~L~(E4XEs;ET) the associated l i n e a r map~.

- 127 -

w

Further by c : E4 x E T - ~ - - ~ E 8 the composition map, and as

before by c 2 : E4 ----~L~(E5IE6) the mapping f ; Finally D = E 5 and id : E 4

~f*.

) E? denotes the map defined by O(f) = f' ) E 4 the identity.

The evaluation map s : Lq(E2;E 3) x E2-----gE 3 is bilinear and continuous by (6.2.3). Hence by (ii.I.I) and (2.9.2) =e have e

L~(E8, E4; E?).

6

Le(E8, E4; E7) ~

By (6.4.11)

=e have

L¢(E8; L~(E4~-?))-

We denote by ~ I the element thus corresponding to e". III' ~2 and id are obviously linear and continuous. Ill, ~r2 ere continuous by (6.3.31 D is linear and continuous by (I0.2.3). c is Cp by our induction hypothesis, c 2 is Cp by (11.2.20). A linear map is obviously Cp. Hence also idxD by (4.4.2). As a consequence of the formula c ' ( f , g ) . ( ~ , ~ ) -

(see (Ii.2o7)) =e have c' = ~

= ~ (g'of,~)+~=f .

6~l=c o(idxO) + ~r 2 oc 2 o W l which

is the composite of Cp-mappinge~ hence (ii.2.22) by (10.4.7). We further assert that (11.2.25)

c

:

E4

x

E5 - -

~

E6 i s C1.

From (11.2.?) me conclude t h a t the mapping c : Ck(EI;E2) x Ck+p+2(E2;E3)----~Ck(EI;E3) is differentiable. The spaces are eli admissibie by (I0.2.4).

-

128

-

The mapping c m = E4 x E5 ----~L(E4xE5;E6) (10.1.4) and ( 2 . 6 . 3 ) . (11.2.8). c'

= E4 x

is Co by (11.2.22),

We have v e r i f i e d the assumptions of

Hence c I E4xE5 ~ E

6 is diffsrenttable

by ( 2 . 6 . 4 ) .

E5 -----~Le(E4xE5;E6) being Co by (11.2.221 and ( 1 0 . 1 . 4 ) ,

=e get (11.2.23). Applying (11.4.1) to (11.2.22) and (11.2.23) one completes the i n d u c t i o n proof of the theorem. (11.2.24)

Latona. Let f: If i k o f

Eo----~ E = proj. lira Ei . t~I

= E° ~ E

f : Eo ~

k is Cp for each k 6 I~ then E is Cp.

Proof. Let p = O; ~VJ~,IE~Eo. Then ~J(ikof)(J~) = ik(~f(J~)) ~ Ek

f o r each k E I . ~f(~,

Hence ~Vf(J~)~ E. S i m i l a r l y one shows that

X ) ~F.

Suppose the lemma for p and assumes ik=f : E° -----~Ek is Cp+ l for each k E I. Then by (10.1.3) (ik=f)' = ikof' : Eo----~L4 (Eo;Ek) is Cp for each k E I. Since the inclusion i k = E

~ Ek is linear and continuous, so

i s ( t k ) . : L(Eo;E)-----~L(Eo;Ek) L~(Eo;E)-----~L~(Eo;Ek)

(11.2.25)

by ( 6 . 3 . 3 ) ,

by ( 2 . 9 . 1 ) .

=s assert

L(Eo,E) = p r o j . lira L ( E o ; E i ) . i£I

and also ( i k ) . :

-

129

-

One first verifies that the underlying sets are the same. We remark however, that the projective system L(Eo;Ei) , i ~

I, is

slightly more general than those considered in 2.3, since the maps Jk! : L(Eo;Ek) ik~: E k

J L(Eo;E|) induced by the inclusions ~ Et

,

k

~

~,

are not inclusions in the strict sense. This implies that the underlying set of proj. lim L(Eo;Ei) is not the intersection of i ~ I the sets L(Eo;Ei) , i ~ [, but has to be constructed in the usual manner. For this one verifies that the maps Jkt

satisfy the

transitivity condition jemOJkt = Jkm and that they are continuous. Both conditions are easily verified, because Jkl = (ik~)*" The structure of proj. lim L(Eo;Ei) is the coarsest for which the i ~ I induced maps Jk = (ik)* : p~oj. lim L(Eo;Ei) i~I tinuous. Furthermore: ~ L ( E o ; E ) , S ik(~(J~)) i k. ~ =

L ~(jq.)~ E for

= (ik.~)(j~) ~ E k for k ~ I and

i L(Eo;E k) are con-

W~Eo-'-'--~. WJ~&Eo~

(ik).(~) ~ L(Eo;Ek) for k ~ I.

This proves that also the structures of L(Eo;E ) and proj. lim i £ I L(Eo;Ei) are the same, hence (11.2.25). By appendix (5), (g.l.1) and (7.4.6) we get Le(Eo;E ) = proj. lim L~(Eo;Ei). i c I Applying the induction hypothesis to i' k- f ' one concludes: f' : E °..... ~Le(Eo;E) is Cp.

: E

o

; L~(

Eo;Ek)

-

130

It remains to show that f: E° ~

-

E is differentiable. We have

by assumption (ikef)'(x) = ikof'(x ) ~ L~(Eo;Ek) and R(ik, f)(x) a R(Eo;Ek) for each k 6 I.

Let @ ~ E o. Then

(ikof'(x)). ~E = ik(f'(x).~)~ Ek for each k • I, hence f'(x).~IE

and thus f'(x) ~ Le(Eo;E).

Let ~ V ~ E o. Then (of. proof of (3.1.5))

e~(ikof)(x).(W,~) = ik(eRf(x).(W,~)) ~ Ek for each k 6 I. It follows that

eRf(x).(~V,~)~ E, hence

Rf(x) & R(Eo;E). Since f' is C o by the induction hypothesis and (10.1.3), the assertion of the lemma is a consequence of (10.4.1).

(11.2.25)

Theorem.

The composition map

c, C~(EI~E 2) ~ c~• (E2;E3) ~

C~(EI~E 3) is c ~ .

Proof. By appendix (5), (10.2.4) and the definition given in I0.3 we have C~(E1;E2)

= proj.lim~, Ck#(EI;E3). The inclusion

~ (E2;E3) is continuous C$.. (EIIE2) x C p (E2;E 3) c Ck~(E 1;E2) x C k+p+l

and thus by (11.2.21) c: C~(E1;E2)xC~(E2;E3)----~Ck ~ (E1;E 3) is C

P

for each p, k 6 h~ °. From the lemma (11.2.21) one concludes

~ ~ c: C~(EI;E 2) x C~(E2;E3)-----~C~(E1;E3) hence the assertion of the theorem.

is Cp for each p e N

o

-

§ 12.

12.1

131

-

DIFFERENTIABLE DEFORMATION OF DIFFERENTIABLE ~APPlNGS.

Th~ differentiability of the evaluation map.

(12.1.i)

Lemma.

The mapping

Ul: Co~OR;E)----~E defined by ul(f) ~ f(I) is linear and continuous. Proof. The llnearity is obvious. Furthermore ~.I~IR, hence the result.

(12.1.~)

Lemma.

The mapping

u: E

~C:~R;E) defined by u(x).~ = ~.x is linear

end continuous. Proof.

By (6.4.2) and (g.l.l) we have E~L~(~;E). Obviously

L$(~;E) c Cee~R;E), and L~OR;E) has the structure induced by its inclusion in Co~(R;E). Hence the result by (2.3.6).

(12.1.~)

Theorem. The evaluation map e: Ck~+I(EI;E2) x E1 -----~E 2 is Ck. Proof.

The composition map

~: CkI'÷I(EI;E2) x Co~POR;EI)----'~Co~(IR;E2) (*) is Ck by (ii.2.21). Hence e = UlO ~ 6 (idx~) is Ck by (12.1.1), (12.1.2) end (10.4.?).

(*)

~(f,g) = c(o,f) = f.g

-

Because t h e i n c l u s i o n

132

CJ(E1;E2)

c

-

C kq+ l ( E 1 ; E 2 )

is continuous

for

each k G R~ o we g e t o b v i o u s l y :

(12.1.4)

Theorem. The e v a l u a t i o n f

e" C,= (E1;E 2) x E1

12.2

The l i n e a r

(12.2.1)

) E2 i s Cw

this

First

~E denote the map defined by

= x+y and ~:

by ~ ( x ) . y Proof.

E---~C~

= x+y. Then ~

we show t h a t ~ ( x )

i s immediate because ~ ( x )

Furthermore ~(x+h) The mapping h ~ - - - ~

- ~(x)

(E;E) t h e map c a r e c t e r i z e d

is C~

.

i s an element o f C ~ ( E I E ) . is a translation

= ~ where ~ ( y )

of E into

i s a c o n s t a n t map f o r of C~(E;E).

•

homeomorphism C ~ ( E I I C ~ ( E 2 . L E 3 ) ) ~ c ; ( E l x E 2 1 E 3 ) .

Lemma. Let s: ExE s(x,y)

map

C~ (E;E)

= h for

is linear

= ~ for

each x •

each x ~ E.

each y = E. and c o n t i n u o u s .

each h • E, hence o b v i o u s l y

T h i s shows t h a t ~ i s d i f f e r e n t i a b l e

and t h a t ~ ' ( x ) . h

for

an element

throughout

E. C o n s e q u e n t l y ~ '

Lemma. The mapping el: Cw(E2;E3)

E

is a cons-

t a n t map of E into L(E;C~ (E;E)) and thus C~ • (12.2.2)

But

~C=.(C==(EI; E2);C~(E1 E3))

defined by cl(g) = g. is linear and continuous.

-

133

-

Proof. The linearity of c I is obvious, g. is an element of ~ E3) C=.(C=.(EI;E2) x C=~(EI;

by (11.2.26) since g, is a partial

mapping of the composition map.V~C~.(E2;E3). We assert: (12.2.3)

~ . ~ Ck(C.. (E1;E2);C ~(E1;E3)) for each k • R~ o.

Let k = O, ~VJ~,~CJ(E1;E2). Then ~.(J~) =~'~:~c(J~xO,Ox~)~ Ca(El;E3) by (11.2.25) Suppose now the assertion for k. By (10.2.3), (10.3.3) and

(2.g.l) we have ~'~C. (E2;Lf(E2;E3)) Fromthe induction hypothesis one deduces

( ~ , ) . ~ Ck~C~ (EI~ E2) ~co(q $ ;L* (E2~E3))>. The evaluation map e: L$(E2IE3 ) x E2-----)E 3 is continuous by (6,3.3). Using (ii.I.I) and (2.9.2) one easily verifies that

e'l C~I(EI;L~(E2;E3)) x Ce(EI;E2))-----~C $ . -

..(E1;E3)

is continuous. The same is true for the corresponding map

el: C..~(EI;L*(E2;E3))

,L (C~.(E1;E2);C~.(EI;E3))

(cf.(6.4.11)). From (10.4.3) and (11.2.7) one concludes that #

(~.)' = (~i).((~').) ~Ck(C~(EI;E2);L$(C~(EI;E2);C=-(El;E3)~Further we had (case k =

0),~.~Co(C£(E1,E2),C.(EI~E3)).

By (10.4.2) it follows that ~. ~Ck+I(C"~(EI;E2)

; C.~ (El ;E3))

which finishes the induction proof of (12.2.3). From (10.3.2) and (2.g.1) one concludes the assertimn of the lemma.

-

(12.2.4)

134

-

Lemma. If f e Co.(El;E2), then

f*

L(c; (E ;E3) ,,.C"(El;E3)) .

Proof. Let ~$C'.~ (E2;E3), then f*(~) =~,f~C~.(EI;E3) by (11.2.26), because ~or =mc(fx0,0x~) and c ie equably continuous.

(12.2.5)

Theorem. There is a natural linear homeomorphism ~: C~ (ElXE2;E 3) ~

C.~(E 1;C. (E2;E3)), the map ~b-

being caracterized by (~g)(xl).X 2 = £ (Xl,X2)" Proof. Let if: E1 jections Xl~

~ElXE 2 resp. i2:E2-----~EIxE2 be the in-

~(xl,O) resp. x2~

~(O,x2).

Then

g(xl,x 2) = g(il(x I) + i2(x2) ) = (g,~(il(Xl))-i2)(x2) where s: (EIXE2) x (EIXE2)

~EIXE 2 is the map discussed in the

lemma (12.2.1). Hence (~g)(xl) = g.s(il(xl))oi2 = (12og,os°~l)(Xl) where i I • E1-----~ElxE2 is linear and continuous, ~." ElXE 2---~C~(EIxE2~EIxE2) is C, by (12.2°i), g.: C~(ElXE2;ElXE2)~ ~C~(EIxE2;E3) is C ~ which follows easily from (Ii.2.21), g. being a partial mapping of the composition map; finally i .2 * • C~ (EIxE2;E 3)

>C~(E2;E3) is Coo for the same reasons.

By (10.4.7) it follows that ~z-g is an element of C.~ (E1 •,C ~ (E2;E3). Let

E4 = Coo (ElXE2; E3) , E5 = C~. (ElXE2;ElXE2), E6 = C..( E l",C~(E2;E3) ).

- L35 -

Then V #

~' (i~). o(~oi 1) * o cI whets cI ( L . tL' - 4.C ..(EasE4) ) by

(12.2.2), ("s . i l ) * ¢ L(C~(Es,E4),C~" (ELSE4)) by ( 1 2 2 4 ) . (i2) . ~ L(C~(EI;E4);E 6) by (10.4.3), (2.9.1) and the same arguments as in the proof of (11.2.26). Hence ~Tr is the composite of linear and continuous maps and thus obviously linear and continuous. Conversely let ~ : C~(E1;C~-~(E2;E3) )

~Ce6- (EIXE2;E3) denote

the linear map defined by (~f)(xl,x2) = f(xl).X 2. f = e o[f.iTl, IT2~

where e: C~..(E2;E3) x E 2 ~ E

Then 3 is the

evaluation map and ~l reap. qt2 are the projections of ElXE 2 into E1 resp. E2.

Hence by (10.4.5) and (10.4.7)

~ f is an

element of C~(EIXE2;E3). We shall prove the continuity of the mapping ~=

~

by the formula

e. o ~os(i2(1T2))oilo IT 1 and using the abbreviations

~6

=

c*-"(ElSe-" *

(E2;E3)) '

E7 = C ~ (EIXE2;C ~ (E2;E3)), E8 = C~, (EIXE2;E2) , E9 = C% (ElXE2;C ~ (E2;E3)xE2). One has ~rI E C~.(E6;E 7) by (12.2.4), iI E C.~(E7~E7xE8), ~(i2(Tf2) ) C C~.(E7xEs;E7xE8) by (12.2.1), c~ ~ C~(E7xE8;Eg) by (10.4.5) and e. ~ C@(E9;C@~(ElXE2;E3) ) by (12.1.4) and (11.2.26).

-

From (I0.4.7) it follows that ~

136

-

is C ~ and thus obviously con-

tinuous. Furthermore ~ o ~

and ~ o ~

which ends the proof.

ere the corresponding identities

-

137

-

APPENDIX (1) Proposition. If E is an admissible vector space, then the scalar multiplication:

IR x E---~E is equably

continuous. Proof. Let X ~ E .

Then by (2.1.i),

(2.4.2) and (7.i.I)

: (Xv(- X))'LE. If Y £ ~

, then Y ) (X•(-X)),

X ~]E. Thus for any xGX the

segment [ - x , x ] ¢ (Xu(-X))" and consequently

~.xC(Xv(-X)) ~

for each #~#il and x (X, hence IIX ~Y where I 1 = [ - I , i ] , therefore 11 )E i ~

LE. By (2.4.2) also

and

~ . I 1 X = I~ X ~ E for

each ~# O. This proves the equable continuity of the scalar multiplication by (2.8.8) and (2.5.1). (2) Proposition. Let ]( be a filter on a vector space. Then the filter X * =

W.sup (~.~()

is the finest one

among all equable filters coarser then Proof. Obviously X'is equable. Let M E ~*. Then m~V.A, V ~ W

'

A ~ ~su

for all

Choose o( ~ V,

~

~(

= ~ (W~)

= (~V)~

where

,

~ ~ is coarser then ~( .

be any equable filter coarser then ~ ~ # 0 :

.

~ O. Since A ¢ ~ (

~.A (~( , and therefore m ( ~

A. This proves that

Then, for • ~

•

~ ~ D, we have

because M ~ ~ Let now ~

(~ X )

~

= W.~ = ~

;

hence

: ~i

~ = I;~

-

~up ( ~ )

" ~

and thus

~"

138

-

= V.sup ( ~ )

which completes the proof. Corollary. For any pseudo-topological vector space E one has:

E

X' E.

(3) Proposition. Let E be any pseudo-topological vector space. Then

( ~ ~ E implies sup ~

E equable ~

Proof. (e) Suppose E equable, and let ~ E . with

~

•

~

= V. ~ LE. Since •

for

~

E).

Then there exists

is coarser then ~ and equable,

~ ~ O; hence su

(~)~

i

(b) Suppose the condition satisfied, and let

~IE.

showing that ~1.~

me have

~

m

~

= W. ~

Then

~E, which proves that E is equable.

(4) Proposition. Let El, i gI be a projective system of equable and admissible vector spaces (cf.(2.3.5)(c)). Then E' = proj.lim E i is equable and admissible. i¢ I Proof. The admissibility is proved in (7.3.2). Let ik(~) L E k for each k g I. But sup ~ ik(~) ~$0

one has

U ~ ~0

ik(X~ ) =

such union. Hence sup ~ ~ E J$O

proposition.

U ~0

~ ~E. Then

: ik(sup ~ ~ ), since 8~0

[k(~X~)

= ik( ~tJ 0 ~ ~

for each

and thus E is equable by the preceding

-

139

-

(5) Proposition. For each projective system of admissible vector spaces, (proj.lim Ei )e = proj.lim £i$. i~l iel Proof. (a) Let )&~(proj.Iim £i)°. Than there is ~ with itl ~( " ~ = W~&proj.lim El" Since ik(K)"ik(~) = ik(V~) = Wik(~), i&I one gets ik(K)~ £kf for each k ~I and thus K&proj.lim £~. l~I i (b) If X proj.lim Eit , then ik(X ) &E k for each k6I by ilI (2.6.3). But by (4) and (2.6.1) we can suppose that ]( is equable, hence ~(proj.lim Ei )$ . i~I

-

140

-

NOTATIONS

~ , $', X

filters

@

empty set

[8],[A],[a]

generated filters

i.I I.I 1.1

comparison of filters

(1.2.1)

][converges to x on E

2.1

~converges to zero on E

{Xi' i.I sup

i~I

~

family of filters

2.4

1.2

least upper bound of filters

1.2

l

XlvK2

sup (~(l' "~2 ) SupX

Xo

1.2

(2.4.3)

X

2.7 2.7

,(xF

(5.3.3)

E, E l, E 2

pseudo topological spaces

i

underlying set

Ee

eouable space associated to E

E°

localiy convex space associated to E

E1 z_ E2

comparison of structures

E1 x E2, x E . i~I I

direct product of pseudo-topological spaces 2.3

2.6 2.7

2.3

-

141

-

direct product of filters ( on a direct

~I x X 2

product of two sets) 1,4 the reels (with the natural topology) neighborhood filter of zero in closed interval in closed interval [ - ~ , ~ ] , ~ •o

Ij

{ 1,2,3,.-- l N°

{0,1,2,...I

proj. lim Ei i~l R(EI;E 2)

projective limit

Ln(EI~E 2)

space of n-linear maps

Ck(EIIE2)

space of Ok-mappings

10.2

C,~(EI ;E2 )

space of C,,-mappings

10.3

L'(EI,E2)

instead of (Ln(EI;E2)) °

(6.1.7)

c:(q E2)

instead of (Ck (E I ;E2)~

(6.1.7)

C~(EI ;E2 )

instead of (C~(EI;E2)) m

f : El,,,

mapping f of E 1 into E 2

xs

set of remainders

m E2

(anonymous) map

fl x f2'

(3.1.2) (6.1.6)

x is sent into y under the considered

;y

[fl,f2 ] ,

2.3

fi

1.3

x f.

1.3

i~l

i~l z

-

IT

k

142

-

k-th projection

c

composition map

e

evaluation map

2.3

map associated to u

(ii.i.i)

U°X

instead of u(X)

~f(x,y)

abbreviation of f(x+y)-f(×)

ef(i,x)

abbreviation of

{~0

~or

~~ 0

for

~

=0

(4.3.2)

f'(a)

differential quotient

f'(a), Df(a)

derivative of f at the point a

Rf(a)

remainder belonging to 8 map f which is differentiable at a

f(k)(a)

see

Dlf(al,e2),D2f(al,a2 )

partial derivatives

f

f*(g) = gof

f.

f. (~) = f-0

,,,,,,@

implies

(5.2.4)

9.2

if and only if linearly homeomorphic

8.I

(3.2.2)

-

143

-

I N D E X

admissible

7.i

almost all

5.1

associated locally convex topological vector space

2.7

canonical isomorphisms

6.4

chain rule

3.3

coarser

2.3

compatible

2.4

composition map

6.3

continuous

2.2

continuous with respect to associated structures

2.9

Ck-mapping

iO.l

derivetlve

3.2

diagonal map

1.3

differentiable at e point

3.2

differentiable map into e direct product

4.4

differential quotient

4.3

direct product

2.3, 4.4

equable continuity

2.8

equable filter

2.5

equable pseudo-tooological vector space

2.6;appendix~

evaluation map

6.2

filter

i.i

filter-basis

i.i

finer

2.3

Fr~chet condition

4.1

function spaces

6.1,6.2

fundamental theorem of calculus

5.1

-

144

-

higher derivatives

g.l

higher order chain rule

g.2, 10.4

homeomorphism

2.2

images of filters under mappings

1.4

inclusion map

2.3

induced structures

2.3

infimum of filters

1.2

infinitely differentieble

g.2

local cerscter

3.4

mappings into direct products

1.3

mesn value theorem

5.2

netursl

6.4

neighborhood, E-neighborhood

3.4

open, E-open

3.4

partial mapping

8.1

partial derivatives

8.1

projective limit

2.3

pseudo-topological space

2.1

pseudo-topological vector space

2.4

pseudo-topology

2.1

quasi-bounded filter

2.5

quasi-bounded map

2.8

remainder

3.1

separated

3.1

subspece

2.3

supremum of filters

1.2

-

145

-

symmetry of f"(×)

9.1

underlying space

2.1

uniform convergence on bounded sets

6.1

-

146

-

REFERENCES

[1]

Bastiani A.

"Applications diff~rmntiables et vari~t@s

:

diff~rentiables de dimension infinie"~ Journal d'Analyse Math~matique XIII (1964) p.l-ll4. [2]

Binz E.

: "Ein Differenzierbarkeitsbegriff in limitLerten Vektorr@umen", Comm. Math. Helv. 41 (to appear).

[3]

Dieudonn~ J. : "Foundations of modern analysis", Academic Press 1960.

[4]

Fischer H.R. : "Limesrgume", Math. Annalen 137 (1959) p.269-303.

Is]

Keller H.H.

: "R~ume stetiger multilinearer Abbildungen als Limesr~ume", Math. Annalen 159 (1965) p.25g-270.

[6]

Keller H.H.

: "Differenzierbarkeit in topologischen Vektorr~umen", Comm. math. Helv. 3B (1964) p.308-320°

[7]

Keller H.H.

:

"Uber Probleme die bei einer Differentialrechnung in topologischen Vektorr@umen auftreten", Nevanlinna Festband, Springer (to ~ppear).

: "Topologische linearB R~ume", Springer 1960.

[8]

K~the G.

[q]

Kowalsky H.J.: "Topologische R~ume"~ Birkh~user 19ill.

Offsetdruck: Julius Belt~ Weinheim/Bergstr.