Introductory Functional Analysis With

Provides avenues for applying functional analysis to the practical study of natural sciences as well as mathematics. Contains worked problems on Hilbert space theory and on Banach spaces and emphasizes concepts, principles, methods and major applications of functional analysis.

Erwin O. Kreyszig was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada. He was a pioneer in the field of applied mathematics: non-wave replicating linear systems.

Metric Spaces

1

(49)

Metric Space

2

(7)

Further Examples of Metric Spaces

9

(8)

Open Set, Closed Set, Neighborhood

17

(8)

Convergence, Cauchy Sequence, Completeness

25

(7)

Examples. Completeness Proofs

32

(9)

Completion of Metric Spaces

41

(8)

Normed Spaces. Banach Spaces

49

(78)

Vector Space

50

(8)

Normed Space. Banach Space

58

(9)

Further Properties of Normed Spaces

67

(5)

Finite Dimensional Normed Spaces and Subspaces

72

(5)

Compactness and Finite Dimension

77

(5)

Linear Operators

82

(9)

Bounded and Continuous Linear Operators

91

(12)

Linear Functionals

103

(8)

Linear Operators and Functionals on Finite Dimensional Spaces

111

(6)

Normed Spaces of Operators. Dual Space

117

(10)

Inner Product Spaces. Hilbert Spaces

127

(82)

Inner Product Space. Hilbert Space

128

(8)

Further Properties of Inner Product Spaces

136

(6)

Orthogonal Complements and Direct Sums

142

(9)

Orthonormal Sets and Sequences

151

(9)

Series Related to Orthonormal Sequences and Sets

160

(7)

Total Orthonormal Sets and Sequences

167

(8)

Legendre, Hermite and Laguerre Polynomials

175

(13)

Representation of Functionals on Hilbert Spaces

188

(7)

Hilbert-Adjoint Operator

195

(6)

Self-Adjoint, Unitary and Normal Operators

201

(8)

Fundamental Theorems for Normed and Banach Spaces

209

(90)

Zorn's Lemma

210

(3)

Hahn-Banach Theorem

213

(5)

Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces

218

(7)

Application to Bounded Linear Functionals on C[a, b]

225

(6)

Adjoint Operator

231

(8)

Reflexive Spaces

239

(7)

Category Theorem. Uniform Boundedness Theorem

246

(10)

Strong and Weak Convergence

256

(7)

Convergence of Sequences of Operators and Functionals

263

(6)

Application to Summability of Sequences

269

(7)

Numerical Integration and Weak* Convergence

276

(9)

Open Mapping Theorem

285

(6)

Closed Linear Operators. Closed Graph Theorem

291

(8)

Further Applications: Banach Fixed Point Theorem

299

(28)

Banach Fixed Point Theorem

299

(8)

Application of Banach's Theorem to Linear Equations

307

(7)

Applications of Banach's Theorem to Differential Equations

314

(5)

Application of Banach's Theorem to Integral Equations

319

(8)

Further Applications: Approximation Theory

327

(36)

Approximation in Normed Spaces

327

(3)

Uniqueness, Strict Convexity

330

(6)

Uniform Approximation

336

(9)

Chebyshev Polynomials

345

(7)

Approximation in Hilbert Space

352

(4)

Splines

356

(7)

Spectral Theory of Linear Operators in Normed Spaces

363

(42)

Spectral Theory in Finite Dimensional Normed Spaces

364

(6)

Basic Concepts

370

(4)

Spectral Properties of Bounded Linear Operators

374

(5)

Further Properties of Resolvent and Spectrum

379

(7)

Use of Complex Analysis in Spectral Theory

386

(8)

Banach Algebras

394

(4)

Further Properties of Banach Algebras

398

(7)

Compact Linear Operators on Normed Spaces and Their Spectrum

405

(54)

Compact Linear Operators on Normed Spaces

405

(7)

Further Properties of Compact Linear Operators

412

(7)

Spectral Properties of Compact Linear Operators on Normed Spaces

419

(9)

Further Spectral Properties of Compact Linear Operators

428

(8)

Operator Equations Involving Compact Linear Operators

436

(6)

Further Theorems of Fredholm Type

442

(9)

Fredholm Alternative

451

(8)

Spectral Theory of Bounded Self-Adjoint Linear Operators

459

(64)

Spectral Properties of Bounded Self-Adjoint Linear Operators

460

(5)

Further Spectral Properties of Bounded Self-Adjoint Linear Operators

465

(4)

Positive Operators

469

(7)

Square Roots of a Positive Operator

476

(4)

Projection Operators

480

(6)

Further Properties of Projections

486

(6)

Spectral Family

492

(5)

Spectral Family of a Bounded Self-Adjoint Linear Operators

497

(8)

Spectral Representation of Bounded Self-Adjoint Linear Operators

505

(7)

Extension of the Spectral Theorem to Continuous Functions

512

(4)

Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator

516

(7)

Unbounded Linear Operators in Hilbert Space

523

(48)

Unbounded Linear Operators and their Hilbert-Adjoint Operators

524

(6)

Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear Operators

530

(5)

Closed Linear Operators and Closures

535

(6)

Spectral Properties of Self-Adjoint Linear Operators

541

(5)

Spectral Representation of Unitary Operators

546

(10)

Spectral Representation of Self-Adjoint Linear Operators

556

(6)

Multiplication Operator and Differentiation Operator

562

(9)

Unbounded Linear Operators in Quantum Mechanics

571

(38)

Basic Ideas. States, Observables, Position Operator

572

(4)

Momentum Operator. Heisenberg Uncertainty Principle

576

(7)

Time-Independent Schrodinger Equation

583

(7)

Hamilton Operator

590

(8)

Time-Dependent Schrodinger Equation

598

(11)

Appendix 1. Some Material for Review and Reference

609

(14)

A1.1 Sets

609

(4)

A1.2 Mappings

613

(4)

A1.3 Families

617

(1)

A1.4 Equivalence Relations

618

(1)

A1.5 Compactness

618

(1)

A1.6 Supremum and Infimum

619

(1)

A1.7 Cauchy Convergence Criterion

620

(2)

A1.8 Groups

622

(1)

Appendix 2. Answers to Odd-Numbered Problems

623

(52)

Appendix 3. References

675

(6)

Index

681

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Introductory Functional Analysis With Applications

Summary

Author Biography

Table of Contents

Supplemental Materials