Linear functional analysis

Itsdevelopment was based on the fundamental observation that the topologicalconcepts of the Euclidean space IRn can be generalized to function spaces aswell.. If, on the other hand, we e

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Linear Functional Analysis

An Application-Oriented Introduction Translated by Robert Nürnberg

ISSN 0172-5939 ISSN 2191-6675 (electronic)

Hans Wilhelm Alt

Technische Universität München

Garching near Munich

Germany

Mathematics Subject Classification: 46N20, 46N40, 46F05, 47B06, 46G10

1985, 1991, 1999, 2002, 2006, 2012,

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Printed on acid-free paper

6

Translation from German language edition:

by Hans Wilhelm Alt

Copyright © 2012, Springer Berlin Heidelberg

Springer Berlin Heidelberg is part of Springer Science + Business Media

All Rights Reserved

Lineare Funktionalanalysis

2016944464

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The present book is the English translation of a previous German edition,also published by Springer Verlag The translation was carried out by RobertN¨urnberg, who also did a marvellous job at detecting errors and mistakes inthe original version In addition, Andrei Iacob revised the English version.The book originated in a series of lectures I gave for the first time at theUniversity of Bochum in 1980, and since then it has been repeatedly used inmany lectures by me and other mathematicians and during this time it haschanged accordingly I provide the reader with an introduction to FunctionalAnalysis as a synthesis of Algebra, Topology, and Analysis, which is thesource for basic definitions which are important for differential equations.The book includes a number of appendices in which special subjects arepresented in more detail Therefore its content is rich enough for a lecturer

to find enough material to fill a course in functional analysis according tohis special interests The text can also be used as an additional source forlectures on partial differential equations or advanced numerical analysis

It must be said that my strategy has been dictated by the desire to offerthe reader an easy and fast access to the main theorems of linear functionalanalysis and, at the same time, to provide complete proofs So there is aseparate appendix where the Lebesgue integral is introduced in a completefunctional analytic way, and an appendix whith details for Sobolev functionswhich complete the proofs of the embedding theorems Therefore the text isself-contained and the reader will benefit from this fact

Parallel to this edition, a revised German version has become available(Lineare Funktionalanalysis, 6 Edition, Springer 2012) with the same math-ematical content This is made possible by a common source text Thereforeone does not have to worry about the content in different versions I amhappy that this book is now accessible to a wider community

If you find any errors or misprints in the text, please point them out tothe author via email: “alt@ma.tum.de” This will help to improve the text ofpossible future editions

I hope that this book is written in the good tradition of functional ysis and will serve its readers well I thank Springer Verlag for making thepublication of this edition possible and for their kind support over manyyears

anal-Technical University Munich, August 2015

H W Alt

V

1 Introduction 1

2 Preliminaries 9

2.1 Scalar product 9

2.3 Orthogonality 11

2.4 Norm 13

2.6 Metric 16

2.8 Examples of metrics 16

2.9 Balls and distance between sets 18

2.10 Open and closed sets 19

2.11 Topology 19

2.14 Comparison of topologies 21

2.15 Comparison of norms 21

2.17 Convergence and continuity 23

2.18 Convergence in metric spaces 24

2.21 Completeness 27

2.22 Banach spaces and Hilbert spaces 27

2.23 Sequence spaces 28

2.24 Completion 30

E2 Exercises 31

E2.6 Completeness of Euclidean space 34

E2.7 Incomplete function space 34

E2.9 Hausdorff distance between sets 35

3 Function spaces 37

3.1 Bounded functions 37

3.2 Continuous functions on compact sets 38

3.3 Continuous functions 39

3.4 Support of a function 41

3.5 Differentiable functions 41

3.7 H¨older continuous functions 44

3.9 Measures 45

3.10 Examples of measures 46

3.11 Measurable functions 47

VII

3.15 Lebesgue spaces 50

3.18 H¨older’s inequality 52

3.19 Majorant criterion in Lp 55

3.20 Minkowski inequality 55

3.21 Fischer-Riesz theorem 55

3.23 Vitali’s convergence theorem 57

3.25 Lebesgue’s general convergence theorem 60

3.27 Sobolev spaces 63

E3 Exercises 66

E3.3 Standard test function 67

E3.4 Lp-norm as p→ ∞ 67

E3.6 Fundamental theorem of calculus 68

A3 Lebesgue’s integral 71

A3.3 Elementary Lebesgue measure 72

A3.4 Outer measure 73

A3.5 Step functions 74

A3.6 Elementary integral 75

A3.8 Lebesgue integrable functions 78

A3.10 Axioms of the Lebesgue integral 79

A3.14 Integrable sets 84

A3.15 Measure extension 87

A3.18 Egorov’s theorem 90

A3.19 Majorant criterion 91

A3.20 Fatou’s lemma 93

A3.21 Dominated convergence theorem 94

4 Subsets of function spaces 95

4.1 Convexity 95

4.3 Projection theorem 96

4.5 Almost orthogonal element 99

4.6 Compactness 100

4.12 Arzel`a-Ascoli theorem (compactness in C0) 106

4.13 Convolution 107

4.14 Dirac sequences 110

4.16 Riesz theorem (compactness in Lp) 112

4.18 Examples of separable spaces 115

4.19 Cut-off function 118

4.20 Partition of unity 118

4.22 Fundamental lemma of calculus of variations 122

4.23 Local approximation of Sobolev functions 122

4.25 Product rule for Sobolev functions 124

4.26 Chain rule for Sobolev functions 125

E4 Exercises 126

E4.4 Strictly convex spaces 128

E4.5 Separation theorem in IRn 129

Table of Contents IX

E4.6 Convex functions 129

E4.7 Characterization of convex functions 131

E4.8 Supporting planes 132

E4.9 Jensen’s inequality 133

E4.11 The space Lp for p < 1 134

E4.13 Compact sets in 2 135

E4.15 Comparison of H¨older spaces 136

E4.16 Compactness with respect to the Hausdorff metric 137

E4.18 Continuous extension 138

E4.19 Dini’s theorem 139

E4.20 Nonapproximability in C0,α 139

E4.21 Compact sets in Lp 139

5 Linear operators 141

5.2 Linear operators 142

5.7 Neumann series 146

5.8 Theorem on invertible operators 147

5.9 Analytic functions of operators 147

5.10 Examples (exponential function) 148

5.12 Hilbert-Schmidt integral operators 149

5.14 Linear differential operators 151

5.17 Distributions (The spaceD(Ω)) 152

5.20 Topology on C∞ 0 (Ω) 156

5.21 The spaceD(Ω) 157

E5 Exercises 160

E5.3 Unique extension of linear maps 160

E5.4 Limit of linear maps 161

6 Linear functionals 163

6.1 Riesz representation theorem 163

6.2 Lax-Milgram theorem 164

6.4 Elliptic boundary value problems 167

6.5 Weak boundary value problems 169

6.6 Existence theorem for the Neumann problem 170

6.7 Poincar´e inequality 171

6.8 Existence theorem for the Dirichlet problem 171

6.10 Variational measure 173

6.11 Radon-Nikod´ym theorem 173

6.12 Dual space of Lp for p <∞ 175

6.14 Hahn-Banach theorem 180

6.15 Hahn-Banach theorem (for linear functionals) 182

6.20 Spaces of additive measures 185

6.21 Spaces of regular measures 185

6.23 Riesz-Radon theorem 187

6.25 Functions of bounded variation 191

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E6 Exercises 193

E6.1 Dual norm on IRn 193

E6.2 Dual space of the cross product 194

E6.3 Integral equation 194

E6.5 Dual space of Cm(I) 195

E6.6 Dual space of c0 and c 198

E6.8 Positive functionals on C0 0 199

E6.9 Functions of bounded variation 201

A6 Results from measure theory 205

A6.1 Jordan decomposition 205

A6.2 Hahn decomposition 206

A6.5 Alexandrov’s lemma 210

A6.7 Luzin’s theorem 212

A6.8 Product measure 213

A6.10 Fubini’s theorem 215

7 Uniform boundedness principle 219

7.1 Baire category theorem 219

7.2 Uniform boundedness principle 220

7.3 Banach-Steinhaus theorem 220

7.7 Open mapping theorem 222

7.8 Inverse mapping theorem 223

7.9 Closed graph theorem 224

E7 Exercises 224

E7.2 Pointwise convergence in L (X; Y ) 225

E7.4 Sesquilinear forms 226

8 Weak convergence 227

8.1 Weak convergence 227

8.2 Embedding into the bidual space 228

8.7 Weak topology 233

8.8 Reflexivity 234

8.12 Separation theorem 240

8.14 Mazur’s lemma 241

8.16 Generalized Poincar´e inequality 242

8.17 Elliptic minimum problem 244

E8 Exercises 249

E8.4 Weak convergence in C0(S) 251

E8.7 Weak convergence of oscillating functions 254

E8.8 Variational inequality 255

A8 Properties of Sobolev functions 258

A8.1 Rellich’s embedding theorem in W0m,p(Ω) 258

A8.2 Lipschitz boundary 259

A8.3 Localization 261

A8.4 Rellich’s embedding theorem in Wm,p(Ω) 261

A8.5 Boundary integral 263

A8.6 Trace theorem 268

A8.8 Weak Gauß’s theorem 270

A8.12 Extension theorem for Sobolev functions 275

A8.13 Embedding theorem onto the boundary 276

A8.14 Weak sequential compactness in L1(μ) 277

A8.15 Vitali-Hahn-Saks theorem 282

9 Finite-dimensional approximation 285

9.3 Schauder basis 288

9.4 Dual basis 288

9.6 Bessel’s inequality 292

9.7 Orthonormal basis 292

9.10 Weierstraß approximation theorem 296

9.13 Linear projections 299

9.14 Continuous projections 300

9.15 Closed complement theorem 301

9.21 Piecewise constant approximation 305

9.22 Continuous piecewise linear approximation 309

9.23 Ritz-Galerkin approximation 311

9.25 C´ea’s lemma 312

E9 Exercises 313

E9.1 Hamel basis 313

E9.2 Discontinuous linear maps 313

E9.8 Projections in L2 ] − π, π[ 316

10 Compact operators 319

10.1 Compact operators 319

10.6 Embedding theorem in H¨older spaces 325

10.7 Sobolev number 327

10.8 Sobolev’s theorem 329

10.9 Embedding theorem in Sobolev spaces 333

10.11 Morrey’s theorem 335

10.13 Embedding theorem of Sobolev spaces into H¨older spaces337 10.14 Inverse Laplace operator 339

10.15 Hilbert-Schmidt integral operator 340

10.16 Schur integral operators 342

10.17 Fundamental solution of the Laplace operator 346

10.18 Singular integral operators 348

10.19 H¨older-Korn-Lichtenstein inequality 349

10.20 Calder´on-Zygmund inequality 351

E10 Exercises 352

E10.2 Ehrling’s lemma 353

E10.8 Sobolev spaces on IRn 356

E10.9 Embedding theorem in IRn 356

E10.10 Poincar´e inequalities 357

E10.13 Nuclear operators 358

E10.15 Bound on the dimension of eigenspaces 359

A10 Calder´on-Zygmund inequality 361

11 Spectrum of compact operators 373

11.6 Fredholm operators 376

11.9 Spectral theorem for compact operators 381

11.11 Fredholm alternative 385

11.12 Finite-dimensional case 385

11.13 Jordan normal form 386

11.14 Real case 387

12 Self-adjoint operators 389

12.1 Adjoint operator 389

12.2 Hilbert adjoint 389

12.4 Annihilator 390

12.6 Schauder’s theorem 391

12.8 Fredholm’s theorem 393

12.9 Normal operators 393

12.12 Spectral theorem for compact normal operators 395

12.14 Eigenvalue problem as a variational problem 397

12.15 Self-adjoint integral operator 399

12.16 Eigenvalue problem for the Laplace operator 400

E12 Exercises 407

E12.1 Adjoint map on C0 407

A12 Elliptic regularity theory 412

A12.2 Friedrichs’ theorem 413

References 419

Symbols 421

Index 425

Functional analysis deals with the structure of function spaces and the ties of continuous mappings between these spaces Linear functional analysis,

proper-in particular, is confined to the analysis of lproper-inear mappproper-ings of this kproper-ind Itsdevelopment was based on the fundamental observation that the topologicalconcepts of the Euclidean space IRn can be generalized to function spaces aswell To this end, functions are interpreted as points in a given space (see thecover page, where a part of the orthonormal system in 9.9 is shown) Given

a set S, we consider the set of all maps f : S → IR Denoting this set by

F (S; IR) means that any point f ∈ F (S; IR) defines a mapping x → f(x)that assigns to each element x∈ S a unique f(x) ∈ IR Then the set F (S; IR)becomes a vector space if we define for all f1, f2, f∈ F (S; IR) and α ∈ IR(f1+ f2)(x) := f1(x) + f2(x) , (αf )(x) := αf (x) for x∈ S With the help of characteristic examples we now investigate similarities anddifferences between the Euclidean space IRn and some function spaces Thefunction spaces will be covered in more detail later on in the book

First we consider the space C0(S) (see 3.2) of continuous functions f :

S → IR, where S is a bounded, closed set in IRn The supremum norm on

x2i

1

for x = (xi)i=1, ,n= (x1, , xn)∈ IRn

One difference between the two spaces is that C0(S), in contrast to IRn, is

an infinite-dimensional space, when S contains infinitely many points Thiscan be seen as follows Let xi ∈ S for i ∈ IN be pairwise distinct Then foreach n ∈ IN we can find functions ϕn,i ∈ C0(S) for i = 1, , n, such that

ϕn,i(xj) = δi,j for i, j = 1, , n Here

denotes the Kronecker symbol Now if αi ∈ IR for i = 1, , n are suchthat

f :=

n

i=1

αiϕn,i= 0 in C0(S) ,

then it follows that 0 = f (xj) = αj for j = 1, , n Hence ϕn,1, , ϕn,nare linearly independent and, since n ∈ IN was chosen arbitrarily, the di-mension of C0(S) cannot be finite This changes the properties of the spacesignificantly For instance, while in IRn all bounded closed sets are compact(see the Heine-Borel theorem 4.7(7)), this is not the case in C0(S) (see theArzel`a-Ascoli theorem 4.12)

Also, the scalar product in IRn,

(x , y)IRn :=

n

i=1

xiyi for x = (xi)i=1, ,n , y = (yi)i=1, ,n∈ IRn,

has an analogue for function spaces; indeed, define (cf 3.16(3))

(f , g)L2 :=

S

f (x)g(x) dx for f, g∈ C0(S)

The corresponding normf L 2 :=

(f , f )L2 is bounded from above by thesupremum norm, that is, there exists a constant C <∞ such that

f L 2 ≤ Cf C 0 for all f∈ C0(S)(this follows from 3.18, if C denotes the square root of the Lebesgue measure

of S) In general, a similar bound from below cannot be derived To see this,consider the interval S = [ − 1, 1] ⊂ IR and for 0 < ε < 1 the functions

fε defined by fε(x) := max

0,1 ε

gk(x) := 1 for x≤ 0, form a Cauchy sequence with respect to the L2-norm,but there exists no function g∈ C0(S) such thatgk− gL2 → 0 as k → ∞

In a situation like this we can apply a general principle in mathematics:completion (see 2.24) Similarly to defining the real numbers IR as the comple-tion of the rational numbers Q, we can complete the space C0(S) with respect

to the L2-norm Thus we obtain the complete space L2(S) of all square tegrable, Lebesgue measurable functions on S (see 3.15 and 4.15(3)) In thisspace fundamental assertions hold, such as Lebesgue’s convergence theorem(see 3.25)

in-We encounter a similar situation in a further generalization from the dimensional case to the infinite-dimensional one For the finite-dimensional

finite-case, let E : S → IR be a continuous function defined on a bounded closedset S ⊂ IRn We now look for a minimum of this function over S Thecompactness of S and the continuity of E yield that such a minimum exists:

E has an absolute minimum on S, that is, there exists an x0∈ S such that

E(x0) = inf

x ∈SE(x) The same holds true if we only assume that S is closed and if in addition werequire that E(x)→ ∞ for x ∈ S as xIR n→ ∞

As an infinite-dimensional analogue we consider the following Dirichletboundary value problem on an open, bounded set Ω⊂ IRn The given datum

is a continuous function u0 defined on the boundary ∂Ω of Ω, i.e u0 ∈

C0(∂Ω), and we want to find a continuous function u : Ω→ IR that is twicecontinuously differentiable in Ω, such that

Δu(x) :=

n

i=1

∂2

∂x2 iu(x) = 0 for x∈ Ω,u(x) = u0(x) for x∈ ∂Ω

In applications, u is, for example, a stationary temperature distribution orthe potential of a charge-free electric field One approach to find a solution

is to consider the corresponding energy functional (here identical to the

Dirichlet integral)

E(u) := 1

2

Ω

Here we use the term functional,

because E acts on functions, that is, E is a function defined on functions Inorder to guarantee that E(u) <∞, we initially define the domain of E to be

M := v∈ C1(Ω) ; v = u0 on ∂Ω

, so E : M→ IR ,where we assume that M is nonempty If we now assume that u∈ M is anabsolute minimum of E on M , then E(u) ≤ E(u + εζ) for all ε ∈ IR andall ζ ∈ C1(Ω) such that ζ = 0 in a neighbourhood of ∂Ω On noting that

ε→ E(u + εζ) is differentiable in ε (this function is quadratic in ε), it followsthat

0 = d

dεE(u + εζ)|ε=0=

Ω

∇ζ(x) • ∇u(x) dx The fact that this identity holds for all functions ζ with the above mentionedproperties contains all the information needed in order to derive a differential

equation for u That is why the functions ζ are also called test functions,

and u∈ M is called a weak solution of the boundary value problem if the

integral identity holds for all test functions Introducing this solution concept

allows the treatment of partial differential equations by means of functionalanalysis (see 6.5–6.8) We obtain the corresponding classical differential equa-tion on assuming that u∈ C2(Ω), as integration by parts then yields that

0 =

Ω

∇ζ(x) • ∇u(x) dx = −

Ωζ(x)Δu(x) dx

for all test functions ζ This implies Δu = 0 in Ω (cf 4.22), and hence u is asolution of the original Dirichlet problem

However, the existence of an absolute minimum u∈ M for a functional

E : M → IR with M ⊂ C1(Ω) is not established as easily as in the dimensional case For instance, if Ω =]0, 1[,

finite-M1:= u∈ C1

[0, 1]; u(0) = 0, u(1) = 1withuC1 :=uC0+uC0 and

E1(u) :=u2

C 0+

 1 0

|u(x)|2

dx ,then M1 is closed in C1

[0, 1] and E1 is continuous with respect to the

C1-norm Moreover, E1(u)≥ u2

C 0 → ∞ for u ∈ M1 asuC1→ ∞, sincefor u∈ M1 and x∈ [0, 1] we have

|u(x)| =

0xu(y) dy

≤ uC0,and henceu2

C 1 ≤ 4 u2

C 0 Consequently, all the assumptions are satisfiedwhich lead in the above finite-dimensional case to the existence of an absoluteminimum

But E1 does not have an absolute minimum on M1 To see this, notethat E1(u)≥ u2

C 0 ≥ |u(1)|2

= 1 for all u∈ M1 This lower bound alsorepresents the infimum of E1 over M1, since the functions u(x) := 1x for > 1 satisfy

... problem and attempt

to solve an infinite-dimensional system of linear equations To this end, let{ei; i∈ IN} be a linearly independent set in the function space

V := v∈ C2... (·1,·2)X is linear in the first argument and conjugate

linear< /b>in the second argument Where no ambiguities arise, one can... x2)X The sesquilinear form is called symmetric (also called a Hermitian form) if for all x, y∈ X one has

A sesquilinear form is called positive semidefinite