Applied Functional Analysis and Partial Differential Equations

Applied Functional Analysis and Partial Differential Equations

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Applied Functional Analysis and Partial Differential Equations

Milan Miklavcic Michigan State University

V f e World Scientific wb

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

First published 1998 Reprinted 2001

APPLIED FUNCTIONAL ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3535-6

Printed in Singapore by Uto-Print

Contents Preface 1

Linear Operators in Banach Spaces 1.1 Metric Spaces 1.2 Vector Spaces 1.3 Banach Spaces 1.4 Linear Operators 1.5 Duals 1.6 Spectrum 1.7 Compact Linear Operators 1.8 Boundary Value Problems for Linear ODEs 1.9 Exercises

2

Linear Operators in Hilbert Spaces 2.1 Orthonormal Sets 2.2 Adjoints 2.3 Accretive Operators 2.4 Weak Solutions 2.5 Example: Constant Coefficient PDEs 2.6 Self-adjoint Operators 2.7 Example: Sturm-Liouville Problem 2.8 Sectorial Forms 2.9 Example: Harmonic Oscillator and Hermite Functions 2.10 Example: Completeness of Bessel Functions 2.11 Example: Finite Element Method 2.12 Friedrichs Extension 2.13 Exercises

47 47 54 58 62 68 70 76 80 88 92 96 99 104

3

Sobolev Spaces 3.1 Introduction 3.2 Fourier Transform 3.3 Distributions

109 109 114 122 v

1 1 6 8 11 16 23 29 39 43

vi

3.4 3.5 3.6 3.7 3.8 3.9

Weak Derivatives Definition and Basic Properties of Sobolev Spaces Imbeddings of Wm*(il) Elliptic Problems Regularity of Weak Solutions Exercises

126 135 140 153 160 163

4

Semigroups of Linear Operators 4.1 Introduction 4.2 Bochner Integral 4.3 Basic Properties of Semigroups 4.4 Example: Wave Equation 4.5 Sectorial Operators and Analytic Semigroups 4.6 Invariant Subspaces 4.7 The Inhomogeneous Problem - Part I 4.8 Exercises

167 167 171 176 187 192 204 209 212

5

Weakly Nonlinear Evolution Equations 5.1 Introduction 5.2 Basic Theory 5.3 Example: Nonlinear Heat Equation 5.4 Approximation for Evolution Equations 5.5 Example: Finite Difference Method 5.6 Example: Galerkin Method for Parabolic Equations 5.7 Example: Galerkin Method for the Wave Equation 5.8 Friedrichs Extension and Galerkin Approximations 5.9 Exercises

215 215 218 222 224 233 236 239 243 245

6

Semilinear Parabolic Equations 6.1 Fractional Powers of Operators 6.2 The Inhomogeneous Problem - Part II 6.3 Global Version 6.4 Main Results 6.5 Example: Navier-Stokes Equations 6.6 Example: A Stability Problem 6.7 Example: A Classical Solution 6.8 Dynamical Systems 6.9 Example: The Chafee-Infante Problem 6.10 Exercises

247 247 254 257 259 267 271 273 276 279 282

vii

Bibliography

285

List of Symbols

289

Index

291

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Preface This book is an introduction to partial differential equations (PDEs) and the relevant functional analysis tools which PDEs require. This material is intended for second year graduate students of mathematics and is based on a course taught at Michigan State University for a number of years. The purpose of the course, and of the book, is to give students a rapid and solid research-oriented foundation in areas of PDEs, like semilinear parabolic equations, that include studies of the stability of fluid flows and, more generally, of the dynamics generated by dissipative systems, numerical PDEs, elliptic and hyperbolic PDEs, and quantum mechanics. In other words, the book gives a complete introduction to and also covers significant portions of the material presented in such classics as Partial Differential Equations by Avner Friedman, Ge­ ometric Theory of Semilinear Parabolic Equations by Dan Henry, and Semigroups of Linear Operators and Applications to PDEs by A. Pazy. The need for such a book is due to the fact that in order to study PDEs one needs to know some functional analysis, which requires a thorough knowledge of real analysis (Lebesgue integral). Therefore, if real analysis is studied in the first year of graduate school, and functional analysis in the second year, the student only begins with PDEs in the third year - and may even have to re-learn functional analysis if the prior instructor ignored unbounded operators (which sometimes happens). The reader is expected to be comfortable with the Lebesgue integral; more specif­ ically, with the material presented in Examples 1.3.4 and 1.5.2. The Cauchy Theorem is also used in a couple of places, with the most difficult version used in (4.44). These are the only real prerequisites for the whole book. Above this level, all theorems used are proved in the text. One may, and perhaps should, skip over some of the proofs. However, they are included in case they are needed. With regard to the writing style, all formal statements, like Theorems, contain all assumptions except for those declared at the beginning of the section in which the statement appears. This should make it easy, even for a casual reader, to figure out what is actually assumed in a given statement. There is, however, one exception. Throughout Chapter 3 it is assumed, unless otherwise specified, that ft is an arbitrary nonempty open set in Mn, n € N. In the first two chapters functional analysis tools are developed and differential operators are studied as examples. Sturm-Liouville operators are nice examples of selfix

X

adjoint operators with compact resolvent and are reused in Chapter 4 as generators of strongly continuous semigroups. Hormander's treatment of weak solutions of constant coefficient PDEs is also presented early on as an example. The foundation of elliptic, parabolic and wave equations, as well as of Galerkin approximations, is given in the section on Sectorial Forms. Throughout the text, completeness of a number of orthonormal systems is proven. The Fourier transform and its applications to constant coefficient PDEs are pre­ sented in Chapter 3. We briefly touch upon distributions and fundamental solutions, and prove the Malgrange-Ehrenpreis Theorem. Most of Chapter 3 is devoted to study of Sobolev spaces. Many sharp results concerning existence and compactness of imbeddings, as well as interpolation inequalities, are proven. These results are applied to elliptic problems in the last two sections. The study of evolution equations begins in Chapter 4 where the semigroup theory is introduced. The Hille-Yosida Theorem for strongly continuous semigroups and Hille's construction of analytic semigroups are presented. The semigroup theory and the results of the previous chapters enable us to discuss linear parabolic and wave equations. In preparation for studies of nonlinear evolution equations, the invariant subspaces associated with the semigroups and the inhomogeneous problem are also examined. A dynamical systems approach to weakly nonlinear evolution equations is given in Chapter 5 with a nonlinear heat equation studied as an example. Trotter's approx­ imation theory is adapted to such equations giving convergence of Galerkin and finite difference type approximations. The chapter on semilinear parabolic equations begins with a very technical section on fractional powers of operators. Our main results contain existence, uniqueness, continuous dependence, maximal interval of existence, stability and instability results. These results are applied to the Navier-Stokes equations, to a stability problem in fluid mechanics, to showing how a classical solution can be obtained, and to the ChafeeInfante problem as an example of a gradient system. I wish to thank S. N. Chow and D. R. Dunninger for their early encouragement and my wife Pam for checking the grammar.

Chapter 1

Linear Operators in Banach Spaces 1.1

Metric Spaces

A metric space is a set M in which a distance (or metric) d is defined, with the following properties: (i) 0 < d(x, y) < oo for all x,y E M (ii) d(x, y) = 0 if and only if x = y (iii) d(x,y) = d(y,x)

for all x,y E M

(iv) d(x,y) < d(x,z) +d(z,y)

for all x,y,z

E M (triangle inequality).

The best known example of a metric space is R n , n G N, with the distance between two points x = (xi,..., xn) and y = ( y i , . . . , yn) given by d(x, y) = \/(xi - yi)2 + • ■ • + (xn -

yn)2.

For the remainder of this section, let M be a metric space with metric d. We shall review basic concepts associated with metric spaces mainly in order to standardize notation and terminology. It is assumed that the reader is somewhat familiar with metric spaces. Hence, our review will be brief. When A C M, B C M are not empty, define dist(A, B) = inf{d(x, y) \ x E A, y E B). Analogously, dist(z, A) = dist({x},A) for x E M. The open ball with center a t x E M and radius r will be denoted by

B(x,r) = {yeM\d{x,y) 1

0 such that B(x,s) C 0. Note that the empty set 0 is open and that the intersection of any finite number of open sets is open. C c M i s said to be closed if its complement Cc = {x G M\x £ C} is open. The closure of a set A will be denoted by A and is defined as the intersection of all closed sets containing A. Note that A is a closed set and if A is closed, then A = A. _ A set A is said to be dense in M if A = M, i.e. A fl B ^ 0 for every nonempty open set B. A metric space is separable if it contains a countable dense set. A set K C M is called compact if from any collection of open sets, whose union contains K, a finite number of sets can be chosen so that their union also contains K. Compact sets are closed in metric spaces. A set is said to be relatively compact if its closure is compact. Recall that a subset of R n , n G N, is compact if and only if it is closed and bounded. A sequence {xn}'^=1 in M is said to converge to x G M if \iuin^00d(xn,x) = 0. Notations xn —> x or limn_>00 xn = x will be used in such a case. Note if A C M, then x G A iff there exists a sequence of points of A converging to x. Observe also that a set K c M i s compact (relatively compact) iff for each sequence {yn} in K there exist y G K (y G M, respectively) and integers n\ < ri2 < • • • such that lim^oo ynk = y. A sequence of elements {xn}^=l in M is said to be a Cauchy sequence if for each e > 0 there exists an integer N such that d(xn,xm) < e whenever n , m > N. A metric space is said to be complete if every Cauchy sequence converges to some element in the space. Theorem 1.1.1 (Baire) If M is a complete metric space, the intersection of every countable collection of dense open subsets of M is dense in M. PROOF Suppose Vi,V2,V3,... are dense and open in M. Let W be any nonempty open set in M. It will be shown that (nVn) H W ^ 0. Since V\ is dense, W D Vi is a nonempty open set. Hence we can find x\ and ri such that B(xu2rx) C WDVx and 0 < n < 1. (1.1) If n > 2 and x n _i and rn-\ are chosen, the denseness of Vn shows that Vn D B ( x n _ i , r n _ i ) is not empty, and we can therefore find xn and rn such that B(xni2rn)cB{xn-Urn-i)nVn

0 < rn < - . (1.2) n By induction, this process produces a sequence {xn} in M. If i > n,j > n the construction shows that Xi and Xj both lie in B(xn,rn). Hence d(xi,Xj) < 2rn < 2/n and therefore {a;n} is a Cauchy sequence. Since M is complete, linin-^oo d(xn, x) = 0 for some x G M. Now, for m > n > 1, d(xn,x)

< d(xn,xm)

and

+ d(xm,x)

< rn + d(x

1.1. METRIC

3

SPACES

and hence d(xn,x) < rn < 2r n , x G B(xn, 2rn). By (1.2), x belongs to each Vn, n > 2. Equation (1.1) implies x G W fl V\. □ Corollary 1.1.2 If M is a complete nonempty metric space and M = U^=1An, some An contains a nonempty open set.

then

If T : M -» M is such that for some e G [0,1) we have that d(T(x),T(y)) < ed(x, y) for all x,y £ M, then T is called a contraction, x G M is said to be a fixed-point of T if T(a;) = x. The following Theorem implies that a contraction mapping on a complete, nonempty metric space always has a unique fixed-point. Try Exercise 2. Theorem 1.1.3 (Contraction Mapping) Suppose M is a complete, nonempty metric space with metric d and that T : M —>• M is such that for some n > 1 and some e < 1 we have d{Tn(z),Tn(y))<ed(z,y)

for all z,y G M.

Then there exists a unique x G M such that T(x) — x. Moreover, d(x,Tm(y))

[—]

< ^—

max

d{Tn+j\y),T3\y))

for all m > 0, y G M.

1 — £ 0 0. Hence, for m > A: > 0 d(xm,xk)

< d(xm,Xm-i) < d(xm, xm-i)

+d(xm-i,xk) + d(z m _i, xm-2) H

m_1

d{xmi xk)

h ^(rcjt+i, rr/j)

fc

< £ d(a:i,xo) + h£ rf(rEi,x0) < £kd(xi,x0)/(l - e).

(1.3)

Thus {xk} forms a Cauchy sequence in M and therefore {xk} converges to some x G M. Since d(Tn(x),xk+i) < £d(x,xk), we see that {xfc} converges also to T n (x) and thus Tn(a;) = x. IiTn(z) = z, then d(x,z) = d{Tn(x),Tn{z))

< ed(x,z).

Hence x = z. Since Tn(T(x)) = T{x) we see that T(x) = x. If T(w) = w, then T n (iy) = w. Hence w = x. Taking m -> oo in (1.3) implies

So, if y is replaced by T J (y), 0 < j < n — 1, the 'moreover' part follows.

□

CHAPTER 1. LINEAR OPERATORS

4

IN BANACH

SPACES

A function / from the metric space M into another metric space N with metric p is said to be continuous at the point x E M if for each e > 0 there exists 8 > 0 such that p(f(x),f(y)) < e whenever y G M and d(x,y) < 8. / i s said to be continuous if it is continuous at each point. / is said to be uniformly continuous if for each e > 0 there exists 8 > 0 such that p(f(x), f{y)) < e whenever d(x, y) < 8. C(M, N) will denote the collection of all continuous functions from M into N. Let CB{M, N) be the collection of those / G C(M, N) for which there exists y G N such that sup t G M p(f(t),y) < oo. Note that d(f,g) = sup p(f(t),g(t)) defines a metric on

^

f,geCB(M,N)

CB(M,N).

Theorem 1.1.4 Let M and N be nonempty metric spaces. (1) If M is compact, then CB{M,N)

=

C{M,N).

(2) If N is complete, then CB{M,N)

is a complete metric space.

PROOF Choose / G C(M,N), y e N and let Ak = {t G M \ p{f(t),y) < k}. Note that Ai, A 2 , . . . are open sets and that M C Ujg^Afc. So, if M is compact then M C Ak for some k and, hence, / G CB(M, N). Assume that N is complete and that {fn} is a Cauchy sequence in CB{M, N). Thus, for each e > 0 there exists k£ such that p(/nW, fm(t))

< e for all t E M, n,m> k£.

(1.4)

Completeness of iV implies that {/n(£)} converges to some f(t) for every t G M . If e > 0 and n > fce/3, then (1.4) implies p(/ n (t), /(£)) < e/3 for £ € M . Hence, p(f(t),y) < e/3 + sup sGM /o(/ n (s),2/) < 00 for some y E N, and since P(/W,/W)


■ W, such that /(0) = x and / ( l ) = y, then W is said to be arcwise-connected. It can be easily shown that if W is arcwise-connected, then it is connected. Theorem 1.1.5 (Arzela-Ascoli) Suppose that M is a separable metric space, with metric d, and that {/ n }^=i is a sequence of complex valued functions on M such that (a) sup n | / n 0&)| < °° for each x G M (b) for each e > 0, x G M there exists 5X£ > 0 such that sup n \fn{x) — fn(y)\ < £ whenever y G M and d(x,y) < 8xe. Then there exist integers n\ < ri2 < • • • and a continuous complex valued function v on M such that lim sup \fnk(x) -v(x)\ =0 for every compact subset K of M. PROOF Let S = {x\,X2,...} be a dense set in M. Let AQ be the set of positive integers. For m > 1 let Am be an infinite subset of Am-i such that lim

j->oojeAm

fj(xm)

exists.

Let no = 1 and for k > 1 choose n^ G A^ such that n^ > n^-i- Note that the lim fni{x)

exists for all x G S.

i-»oo

Let K be a compact subset of M and choose e > 0. Open balls ^ ( a ; , ^ ) with x E K " cover K. Hence, a finite subcollection, say B\,..., Bm, also covers K. Choose yi E S C\Bi and let JV be such that \fni{yk)-fnj(yk)\<e

for

i,j>N,k

Any x G K is contained in some Bk = B(zk,SZk£). \fni(x) - fnj(x)\

=

l,...,m.

Hence for i, j > N


0. If d(x, y) < 5X£, then \v(x)-v(y)\


0, let C m (ft) be the collection of all / G C(Q) such that Daf G C(fi) for every multi-index a with |a| < m. Recall that if / G C m (fi) and a is a multi-index with \a\ < m, then all partial derivatives of / , such that for each i the total number of differentiations with respect to xi is equal to a i , exist and are equal to Daf.

8

CHAPTER

1. LINEAR

OPERATORS

IN BANACH

SPACES

We write / € C°°(ft) iff / <E C m (H) for all m > 0. For m > 0, define C0m(ft) = Co(H) n C m (H) and let C0°°(n) = CQ(Q) n C8°(fi) is often denoted by £>(fi) with its elements called test functions. Cm(ft),

C°°(ft).

C°°(ft), C ^ ( n ) , CS°(ft) are subspaces of the vector space (7(17).

When Q = (a,b) we write C§?(a,b) in place of Cg°((o,6)) ... For a multi-index a = ( a 1 ? . . . , an) we define a! = ax\ ■ • • a n !. If /3 is also a multiindex, we write p 1, m > 0 and define Cjj (ft) to be the set of those / G C m (ft) for which H/llm.oo = max sup|L> a /| < oo. \oc\<m Q

CJf (ft) is clearly a normed space with the norm |H|m,oo- Let us show that Cg^ft) is com­ plete and hence a Banach space. The case m = 0 is discussed above in Example 1.3.2. Assume that CJ l ~ 1 (ft) is complete for some m > 1 and that {fk}kLi is a Cauchy se­ quence in C%(Q). Since { / i b } ^ , {Difk)kLi are Cauchy sequences in C ^ _ 1 ( f t ) , there

10

CHAPTER

1. LINEAR

OPERATORS

IN BANACH

SPACES

exist f,gi G C% 1(Q) such that ||/fc - /|| m -i,oo -+ 0 and \\Difk - #i|| m -i,oo -> 0 as k ->• oo. Hence, if x G ft and e» is the unit vector in E n in the z-th direction, then fk(x for small enough h eR.

+ hei)-fk{x)=

Jo

(Difk)(x

+ se^ds

Letting k —> oo gives f(x + hei) - f(x) =

gi(x + sei)ds. Jo

Hence Dif = gi, which implies completeness. C™(fl) denotes the set of those / G C^(ft) for which Daf is uniformly continuous for every multi-index a with \a\ < m. It can be easily seen that C™(ft) is a closed subspace of C^ift) and hence also a Banach space with the norm || • || m i 0 0 . Note if / € C™(fy, then each Daf, \a\ < m, has a unique uniformly continuous extension on ft which is defined for x G Q\Q by (Daf)(x) = lim n _> 00 ( J D a /)(:c n ) where xn e ft are such that xn -► x. C™{fl) is sometimes denoted by Cm(ty (note that Cm(R) + C m ( E ) ) ; when ft is a bounded interval (a, b) it will be denoted by Cm[a, b]. Let Ct denote the collection of / G C ( E n ) such that limja-i^oo J{x) exists. The set of all / G Ce, such that lim^^QQ f (x) = 0, will be denoted by CIQ. Observe that Ct and CIQ are closed subspaces of C#(R n ) and, hence, Banach spaces. EXAMPLE 1.3.4 Let ft be a nonempty (Lebesgue) measurable set in Rn. For p G [1, oo), we denote by Lp(ft) the set of equivalence classes of measurable functions on ft for which

-a

l/N

Q

0, then CQ-(I) is dense in LP(I), p G [l,oo). If true for some m > 0 and / G LP(I) for some p G [1, oo) and e > 0, then ||/ - g\\p < e/2 for some g G CQ'(I). Define i

rx+8

hs(x) = ^ /

g(t)dt

and note that h6 G CJ n + 1 (7) for small S. Since

hs(x) - g(x) = i j

(g(t) - g(x))dt,

1.4.

LINEAR

OPERATORS

11

the uniform continuity of g implies \\hs - p||o,oo -> 0 as 6 ->• 0, and since the support of hs lies in a fixed compact subset of I for small S we can choose small 8 > 0 such that 11^5 _ 9\\p < £/2- Thus ||/ — hs\\p < e, showing that the statement is true for m + 1 and hence for all m. When p,q,r e [1, oo], p _ 1 + g ~ 1 = r " 1 , Holder's inequality implies that if / G L p (fJ) and p € Lq(n), then / # 6 L r (H) and ll/Pllr < ||/||plMl fl . This implies that if r , p i , ...,p fc G [l,oo], r gi---gke Lr(Q) and

_1

1

= pj" 4-

(1-9) ^P*

1 an

Pi

d #i G L (fi), then

lb---^||roo

F = {x\ \\x\\p = ( E ~ x K l p ) 1 / P < oo},

1 < p < 00.

£°°,co,£.p are nice, easy-to-play-with Banach spaces. EXAMPLE 1.3.6 The Cartesian product I x 7 i s the set of all ordered pairs (x,y), where x G X and y G Y\ If X and Y are vector spaces with the same scalar field, then X x Y is a vector space under the following operations of addition and scalar multiplication: (xi,yi)

+ (£2,2/2) = (zi +£2,2/1 +2/2) a(z,2/) = (ax, ay).

If in addition, X and Y are normed spaces with respective norms || • ||x, 11 • ||y, then X x Y is a normed space with the norm \\(x,y)\\ = \\x\\x + \\y\\y. Moreover, under this norm, X x Y is a Banach space provided X and Y are Banach spaces.

1.4

Linear Operators

Let X , Y b e normed spaces with t h e same scalar field. A m a p p i n g T from a subspace of X, called t h e d o m a i n of T a n d denoted by £>(T), into Y is said to b e a l i n e a r o p e r a t o r f r o m X t o Y if T{ax

+ /fy) = a T z + (3Ty

12

CHAPTER 1. LINEAR OPERATORS

IN BANACH

SPACES

for all x , | / E D(T) and all scalars a,/3. Note that Tx rather than T(x) is used when T is a linear operator. The range, #(T), of T is the set of all Tx with i 6 ! D ( T ) . The null space, K(T), of T is the set of all x G V(T) such that Tx = 0. Thus, a linear operator T is one-to-one iff K(T) = {0}. T is said to be densely defined if T>(T) is dense in X. If D(T) = X, then T is said to be defined on X. If Y = X, then T is said to be defined in X. If S is another linear operator from X to Y, such that V(S) D D(T) and 5a; = Tx for x € D(T), then 5 is called an extension of T and T is called a restriction of S. If S and T are linear operators from X to Y, then the domain of T + S is £>(T) fl X>(5), and if i^ is a linear operator from 7 to Z, then the domain of RS consists of those x EfD(S) for which Sx G T)(R). A mapping T : £>(T) C X —> Y is said to be bounded if it maps bounded sets into bounded sets. When T is linear, T is bounded iff the norm of T defined by \\T\\=Suv{\\Tx\\\xZl){T),h\\(T). The smallest of such m is equal to ||T|| when T is bounded. EXAMPLE 1.4.1 Suppose i C f , B C T and t : B x A -> C are measurable, and that for some Mi, M2 G (0,00) we have that / \t(x,y)\dy < Mi for x £ B , JA

/ |t(x,3/)|da; < M2 for 3/ G A.

JB

Fix p G [1,00] and define an integral operator T : LP(A) -> L p (£) by (Tf)(x) = j t(x,y)f(y)dy

for / G LP(A), z G 5 .

T is a bounded linear operator and ||T|| < Ml/qM2'p where q G [l,oo] is such that 1/q + 1/p = 1. To see this, when p G (1,00), note that Holder's inequality implies |^t(x,i/)/(y)dy| P


0 such that \\Tx\\ < 1 for x G D(T) with ||:z|| < 8. Hence ||T|| < 1/8. If T is bounded, then \\Tx-Ty\\ < \\T\\\\x — y\\ implies continuity. □

1.4. LINEAR

OPERATORS

13

Theorem 1.4.3 Let T : D(T) C X —> Y be a densely defined bounded linear operator from a normed space X to a Banach space Y. Then there exists a unique continuous extension, say Te, ofT on X. Moreover, Te is linear, bounded and \\T\\ = \\Te\\. PROOF Since T is densely defined, for each x e X there is a sequence {xn} C D(T) with xn —y x. Since T is bounded, {Txn} is a Cauchy sequence in y , and thus converges to some y E Y. Moreover, it is easy to show that y is independent of the sequence used. Thus we can (and to obtain a continuous extension of T must) define an operator T e : X —> Y by Tex = lim n ^oo Txn = y. Te is clearly a linear extension of T and thus (1.11) implies ||T e || > ||T||. Since \\Tex\\ = lim ||Ts„|| < \\T\\\\x\\, n—>oo

it follows that \\Te|| = ||T||.

D

The collection of all bounded linear operators T from X to Y with D(T) = X will be denoted by *8(X,Y); define also *8(X) = *B{X,X). Observe that ||TS||