Linear Functional Analysis: An Application-Oriented Introduction

✍ Scribed by Hans Wilhelm Wilhelm Alt

• Publisher: Springer
• Year: 2016
• Tongue: English
• Leaves: 446
• Edition: 1st ed. 2016
• Category: Library

✦ Synopsis

This book gives an introduction to Linear Functional Analysis, which is a synthesis of algebra, topology, and analysis. In addition to the basic theory it explains operator theory, distributions, Sobolev spaces, and many other things. The text is self-contained and includes all proofs, as well as many exercises, most of them with solutions. Moreover, there are a number of appendices, for example on Lebesgue integration theory.

A complete introduction to the subject, Linear Functional Analysis will be particularly useful to readers who want to quickly get to the key statements and who are interested in applications to differential equations.

✦ Table of Contents

Preface
Table of Contents
1 Introduction
2 Preliminaries
2.1 Scalar product
2.2 Lemma
2.3 Orthogonality
2.4 Norm
2.5 Example
2.6 Metric
2.7 Fréchet metric
2.8 Examples of metrics
2.9 Balls and distance between sets
2.10 Open and closed sets
2.11 Topology
2.12 Proposition
2.13 Definition
2.14 Comparison of topologies
2.15 Comparison of norms
2.16 Examples
2.17 Convergence and continuity
2.18 Convergence in metric spaces
2.19 Note
2.20 Note (Nested limits)
2.21 Completeness
2.22 Banach spaces and Hilbert spaces
2.23 Sequence spaces
2.24 Completion
E2 Exercises
E2.1 Open and closed sets
E2.2 Distance and neighbourhoods
E2.3 Construction of metrics
E2.4 Convergence
E2.5 Examples of continuous maps
E2.6 Completeness of Euclidean space
E2.7 Incomplete function space
E2.8 On completeness
E2.9 Hausdorff distance between sets
3 Function spaces
3.1 Bounded functions
3.2 Continuous functions on compact sets
3.3 Continuous functions
3.4 Support of a function
3.5 Differentiable functions
3.6 Space of differentiable functions
3.7 Hölder continuous functions
3.8 Infinitely differentiable functions
Measures and Integrals
3.9 Measures
3.10 Examples of measures
3.11 Measurable functions
3.12 Lemma
3.13 Space of measurable functions
3.14 Theorem
3.15 Lebesgue spaces
3.16 Theorem
3.17 Lemma
3.18 Lemma (Hölder’s inequality)
3.19 Lemma (Majorant criterion)
3.20 Lemma (Minkowski inequality)
3.21 Fischer-Riesz theorem
3.22 Lemma
3.23 Vitali’s convergence theorem
3.24 Corollary
3.25 Lebesgue’s general convergence theorem
3.26 Lemma
3.27 Sobolev spaces
3.28 Theorem
3.29 Wm,p0 (Ω)-spaces
E3 Exercises
E3.1 On uniform convergence
E3.2 Exhaustion property
E3.3 A test function.
E3.4 Lp-norm as p → ∞
E3.5 Subsequences
E3.6 Fundamental theorem of calculus
E3.7 Leftand right-hand limit
E3.8 Estimating the Hölder norm by the W1,p-norm
A3 Lebesgue’s integral
A3.1 Assumptions
A3.2 Consequences
A3.3 Example (Elementary Lebesgue measure)
A3.4 Definition (Outer measure and null sets)
A3.5 Step functions
A3.6 Elementary integral
Construction of Lebesgue’s integral
A3.7 Lemma
A3.8 Lebesgue integrable functions
A3.9 Lebesgue integral
A3.10 Theorem (Axioms of the Lebesgue integral)
Extension of measures
A3.11 Lemma
A3.12 Conclusions
A3.13 Remark
A3.14 Integrable sets
A3.15 Measure extension
Properties of Lebesgue’s integral
A3.16 Measurable functions (see 3.11)
A3.17 Lemma
A3.18 Egorov’s theorem
A3.19 Theorem
A3.20 Fatou’s lemma
A3.21 Dominated convergence theorem (Lebesgue’s convergence theorem)
4 Subsets of function spaces
Convex subsets
4.1 Convex sets
4.2 Convex functions
4.3 Projection theorem
4.4 Remark
4.5 Almost orthogonal element
Compact subsets
4.6 Compactness
4.7 Remarks
4.8 Lemma
4.9 Lemma
4.10 Theorem
4.11 Lemma
Compact sets of function spaces
4.12 Arzelà-Ascoli theorem
4.13 Convolution
4.14 Dirac sequence
4.15 Theorem
4.16 Theorem (M. Riesz)
Dense subsets
4.17 Separable sets
4.18 Examples of separable spaces
4.19 Cut-off function
4.20 Partition of unity
4.21 Examples of partition of unity
4.22 Fundamental lemma of calculus of variations
4.23 Local approximation of Sobolev functions
4.24 Theorem
4.25 Product rule for Sobolev functions
4.26 Chain rule for Sobolev functions
4.27 Lemma
E4 Exercises
E4.1 Subsets of C0 and L1
E4.2 Convex sets
E4.3 Distance in a Banach space
E4.4 Strictly convex spaces
E4.5 Separation theorem in IRn
E4.6 Convex functions
E4.7 Characterization of convex functions
E4.8 Supporting planes
E4.9 Jensen’s inequality
E4.10 Lp-inequalities
E4.11 The space Lp for p < 1
E4.12 Cross product of normed vector spaces
E4.13 Compact sets in l2
E4.14 Bounded and compact sets in L1(]0, 1[)
E4.15 Comparison of H¨older spaces
E4.16 Compactness with respect to the Hausdorff metric
E4.17 Uniform continuity
E4.18 Continuous extension
E4.19 Dini’s theorem
E4.20 Nonapproximability in the space C0,α
E4.21 Compact subsets of Lp
5 Linear operators
5.1 Lemma
5.2 Linear operators
5.3 Theorem
5.4 Remarks
5.5 Definitions
5.6 Examples
5.7 Neumann series
5.8 Theorem on invertible operators
5.9 Analytic functions of operators
5.10 Examples
5.11 Theorem
5.12 Hilbert-Schmidt integral operators
5.13 Definition
5.14 Linear differential operators
Distributions
5.15 Notation
5.16 Lemma
5.17 Distributions
5.18 Examples
5.19 Approximation of distributions
5.20 Topology on C∞0 (Ω)
5.21 The space D(Ω)
5.22 Lemma
5.23 The dual space of D(Ω)
E5 Exercises
E5.1 Commutator
E5.2 Nonexistence of the inverse
E5.3 Unique extension of linear maps
E5.4 Limit of linear maps
E5.5 Pointwise convergence of operators
E5.6 Convergence of operators
6 Linear functionals
Lax-Milgram’s theorem
6.1 Riesz representation theorem
6.2 Lax-Milgram theorem
6.3 Consequences
6.4 Elliptic boundary value problems
6.5 Weak boundary value problems
6.6 Existence theorem for the Neumann problem
6.7 Poincaré inequality
6.8 Existence theorem for the Dirichlet problem
6.9 Remark (Regularity of the solution)
Radon-Nikodỳm’s theorem
6.10 Definition (Variational measure)
6.11 Radon-Nikodỳm theorem
6.12 Theorem (Dual space of Lp for p < ∞)
6.13 Corollary
Hahn-Banach’s theorem
6.14 Hahn-Banach theorem
6.15 Hahn-Banach theorem (for linear functionals)
6.16 Theorem
6.17 Corollaries
6.18 Remark
Riesz-Radon’s theorem
6.19 Definition (Borel sets)
6.20 Spaces of additive measures
6.21 Spaces of regular measures
6.22 Integral of continuous functions (Riemann integral)
6.23 Riesz-Radon theorem (Dual space of C0)
6.24 Corollary
6.25 Functions of bounded variation
6.26 Theorem
E6 Exercises
E6.1 Dual norm on IRn
E6.2 Dual space of the cross product
E6.3 Integral equation
E6.4 Examples of elements from C0([0, 1])'
E6.5 Dual space of Cm(I)
E6.6 Dual space of c0 and c
E6.7 Characterization of Sobolev functions
E6.8 Positive functionals on C00
E6.9 Functions of bounded variation
E6.10 Representation of the Riemann-Stieltjes integral
E6.11 Normalized BV functions
A6 Results from measure theory
A6.1 Jordan decomposition
A6.2 Hahn decomposition
A6.3 Lemma
A6.4 Corollary
A6.5 Lemma (Alexandrov)
A6.6 Lemma
A6.7 Luzin’s theorem
A6.8 Product measure
A6.9 Lemma
A6.10 Fubini’s theorem
A6.11 Remark on the case p = ∞
7 Uniform boundedness principle
7.1 Baire’s category theorem
7.2 Theorem (Uniform boundedness principle)
7.3 Banach-Steinhaus theorem
7.4 Notation
7.5 Theorem
7.6 Definition
7.7 Open mapping theorem
7.8 Inverse mapping theorem
7.9 Closed graph theorem
E7 Exercises
E7.1 On the adjoint map
E7.2 Pointwise convergence in L(X; Y )
E7.3 Equivalent norms
E7.4 Sesquilinear forms
8 Weak convergence
8.1 Definition (weak convergence)
8.2 Embedding into the bidual space
8.3 Remarks
8.4 Examples
8.5 Theorem
8.6 Examples
8.7 Weak topology
Reflexive spaces
8.8 Reflexivity
8.9 Lemma
8.10 Theorem
8.11 Examples of reflexive spaces
Minkowski’s functional
8.12 Separation theorem
8.13 Theorem
8.14 Mazur’s lemma
8.15 Theorem
Variational methods
8.16 Generalized Poincaré inequality
8.17 Elliptic minimum problem
8.18 Examples of minimum problems
E8 Exercises
E8.1 Weak limit in Lp(μ)
E8.2 Weak limit of a product
E8.3 Weak limit of a product
E8.4 Weak convergence in C0
E8.5 Strong convergence in Hilbert spaces
E8.6 Strong convergence in Lp spaces
E8.7 Weak convergence of oscillating functions
E8.8 Variational inequality
E8.9 A fundamental lemma
A8 Properties of Sobolev functions
A8.1 Rellich’s embedding theorem
A8.2 Lipschitz boundary
A8.3 Localization
A8.4 Rellich’s embedding theorem
A8.5 Boundary integral
A8.6 Trace theorem
A8.7 Lemma
A8.8 Weak Gauß’s theorem (Weak divergence theorem)
A8.9 Lemma
A8.10 Lemma
A8.11 Remark
A8.12 Extension theorem
A8.13 Embedding theorem onto the boundary
A8.14 Weak sequential compactness in L1(μ)
A8.15 Theorem (Vitali-Hahn-Saks)
9 Finite-dimensional approximation
9.1 Lemma
9.2 On the Hahn-Banach theorem
9.3 Definition (Schauder basis)
9.4 Theorem (Dual basis)
Orthogonal systems
9.5 Definition
9.6 Bessel’s inequality
9.7 Orthonormal basis
9.8 Theorem
9.9 Example
9.10 Weierstraß approximation theorem
9.11 Lemma
Projection
9.12 Lemma
9.13 Linear projections
9.14 Continuous projections
9.15 Closed complement theorem
9.16 Theorem
9.17 Lemma
9.18 Lemma
9.19 Remark
9.20 Examples
9.21 Piecewise constant approximation
9.22 Continuous piecewise linear approximation
Ritz-Galerkin approximation
9.23 Ritz-Galerkin approximation
9.24 Remark
9.25 Céa’s lemma
E9 Exercises
E9.1 Hamel basis
E9.2 Discontinuous linear maps
E9.3 Dual basis
E9.4 Orthogonal system
E9.5 Weak convergence of unit vectors
E9.6 On the convergence of the Fourier coefficients
E9.7 Projections in a Banach space
E9.8 Projections in L2(] − π, π[ )
E9.9 Projections in a Hilbert space
E9.10 Schauder basis
10 Compact operators
10.1 Compact operators
10.2 Lemma
10.3 Lemma
Embedding theorems
10.4 Lemma
10.5 Theorem
10.6 Embedding theorem in Hölder spaces
10.7 Sobolev number
10.8 Theorem (Sobolev)
10.9 Embedding theorem in Sobolev spaces
10.10 Theorem
10.11 Theorem (Morrey)
10.12 Remarks
10.13 Embedding theorem of Sobolev spaces into Hölder spaces
Laplace operator
10.14 Inverse Laplace operator
Integral operators
10.15 Hilbert-Schmidt integral operator
10.16 Schur integral operators
The fundamental solution
10.17 Fundamental solution of the Laplace operator
10.18 Singular integral operators
10.19 Hölder-Korn-Lichtenstein inequality
10.20 Calderón-Zygmund inequality
E10 Exercises
E10.1 Counterexample to embedding theorems
E10.2 Ehrling’s lemma
E10.3 Application of Ehrling’s lemma
E10.4 On Ehrling’s lemma
E10.5 An a priori estimate
E10.6 Equivalent norm
E10.7 Counterexample to embedding theorems
E10.8 Sobolev spaces on IRn
E10.9 Embedding theorem
E10.10 Poincaré inequalities
E10.11 Convergence in Lp-spaces
E10.12 Compact sets in c0
E10.13 Nuclear operators
E10.14 Compact operator without eigenvalues
E10.15 Bound on the dimension of eigenspaces
E10.16 Norm of Hilbert-Schmidt operators
A10 Calderón-Zygmund inequality
A10.1 Definition
A10.2 Theorem
A10.3 Lemma
A10.4 Theorem
11 Spectrum of compact operators
11.1 Spectrum
11.2 Remarks
11.3 Theorem
11.4 Theorem
11.5 Remarks
11.6 Fredholm operators
11.7 Example
11.8 Theorem
11.9 Spectral theorem for compact operators (Riesz-Schauder)
11.10 Corollary
11.11 Fredholm alternative
11.12 Finite-dimensional case
11.13 Jordan normal form
11.14 Real case
12 Self-adjoint operators
12.1 Adjoint operator
12.2 Hilbert adjoint
12.3 Algebraic properties
12.4 Annihilator
12.5 Theorem
12.6 Theorem (Schauder)
12.7 Remark
12.8 Theorem (Fredholm)
12.9 Normal operators
12.10 Lemma
12.11 Example
12.12 Spectral theorem for compact normal operators
12.13 Remark
12.14 Eigenvalue problem as a variational problem
12.15 Self-adjoint integral operator
12.16 Eigenvalue problem for the Laplace operator
12.17 Theorem
E12 Exercises
E12.1 Adjoint map on C0
E12.2
E12.3
E12.4
E12.5
A12 Elliptic regularity theory
A12.1 Lemma
A12.2 Friedrichs’ theorem
A12.3 Theorem
References
Symbols
Index