**Purpose of the book.**

Functional analysis plays an increasing role within the applied sciences likewise in arithmetic itself. Consequently, it becomes a lot of fascinating to introduce the scholar to the field/sphere at an early stage of study. This book is meant to familiarize the reader with the essential ideas, principles and methods of functional analysis and its applications. Since a textbook ought to be written for the scholar, I have sought to bring basic parts of the field and related practical problems within the comfortable grasp of senior undergraduate students or starting graduate students of arithmetic and physics. I hope that graduate engineering students may exploit the presentation.

**Prerequisites**.

The book is elementary. A background in collegian arithmetic, in particular, linear algebra and ordinary calculus, is sufficient as a prerequisite.

Measure theory is neither assumed nor discussed. No knowledge in topology is required; the few considerations involving compactness are self-contained.

Complex analysis isn't required, except in one of the later sections (Sec.7.5), that is ex gratia, so that it can easily be omitted.

Further assistance is given in Appendix one, which contains simple material for review and reference.

The book ought to so be accessible to a good spectrum of scholars and should conjointly facilitate the transition between algebra and advanced practical analysis.

**Courses.**

The book is appropriate for a one-semester course meeting 5 hours per week or for a two-semester course meeting 3 hours per week.

The book also can be utilised for shorter courses.

In fact, chapters will be omitted while not destroying the continuity or creating the remainder of the book a body part (for details see below).

For instance: Chapters one to four or five makes a awfully short course.

Chapters one to four and seven could be a course that has spectral theory and different topics.

CONTENTS

**Chapter 1. Metric Spaces . . . .**

1.1 Metric Space 2

1.2 Further Examples of Metric Spaces

1.3 Open Set, Closed Set, Neighborhood

1.4 Convergence, Cauchy Sequence, Completeness

1.5 Examples. Completeness Proofs

1.6 Completion of Metric Spaces

**Chapter 2. Normed Spaces. Banach Spaces. . . . .**

2.1 Vector Space 50 2.2 Normed Space. Banach Space 58

2.3 Further Properties of Normed Spaces 67

2.4 Finite Dimensional Normed Spaces and Subspaces 72

2.5 Compactness and Finite Dimension 77

2.6 Linear Operators 82

2.7 Bounded and Continuous Linear Operators 91

2.8 Linear Functionals 103

2.9 Linear Operators and Functionals on Finite Dimensional Spaces 111

2.10 Normed Spaces of Operators. Dual Space 117

**Chapter 3. Inner Product Spaces. Hilbert Spaces. . .127**

3.1 Inner Product Space. Hilbert Space 128

3.2 Further Properties of Inner Product Spaces 136

3.3 Orthogonal Complements and Direct Sums 142

3.4 Orthonormal Sets and Sequences 151

3.5 Series Related to Orthonormal Sequences and Sets 160

3.6 Total Orthonormal Sets and Sequences 167

3.7 Legendre, Hermite and Laguerre Polynomials 175

3.8 Representation of Functionals on Hilbert Spaces 188

3.9 Hilbert-Adjoint Operator 195

3.10 Self-Adjoint, Unitary and Normal Operators 201

Chapter 4. Fundamental Theorems for Normed and Banach Spaces. . . . . . . . . . . 209

4.1 Zorn's Lemma 210

4.2 Hahn-Banach Theorem 213

4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces 218

4.4 Application to Linear Functionals on C[a, b] 225

4.5 Adjoint Operator 231 4.6 Reflexive Spaces 239

4.7 Category Theorem. Uniform Boundedness Theorem 246

4.8 Strong and Weak Convergence 256

4.9 Convergence of Sequences of Operators and Functionals 263

4.10 Application to Summability of Sequences 269

4.11 Numerical Integration and Weak* Convergence 276

4.12 Open Mapping Theorem 285 4.13 Closed Linear Operators. Closed Graph Theorem 291

**Chapter 5. Further Applications: Banach Fixed Point Theorem .**. . . . . . . . . . . 299

5.1 Banach Fixed Point Theorem 299

5.2 Application of Banach's Theorem to Linear Equations 307

5.3 Applications of Banach's Theorem to Differential Equations 314

5.4 Application of Banach's Theorem to Integral Equations 319

**Chapter 6. Further Applications: Approximation Theory**..... . . . . . . . 327

6.1 Approximation in Normed Spaces 327

6.2 Uniqueness, Strict Convexity 330 6.3 Uniform Approximation 336

6.4 Chebyshev Polynomials 345

6.5 Approximation in Hilbert Space 352

6.6 Splines 356

**Chapter 7. Spectral Theory of Linear Operators in Normed, Spaces . .**. . . . . . . . . 363

7.1 Spectral Theory in Finite Dimensional Normed areas 364

7.2 Basic Concepts 370

Contents

7.3 Spectral Properties of Bounded Linear Operators 374

7.4 Further Properties of Resolvent and Spectrum 379

7.5 Use of advanced Analysis in Spectral Theory 386

\7.6 Banach Algebras 394

7.7 Further Properties of Banach Algebras 398

Chapter 8.

Compact Linear Operators on Normed areas and Their Spectrum . 405

8.1 Compact Linear Operators on Normed Spaces 405

8.2 Further Properties of Compact Linear Operators 412

8.3 Spectral Properties of Compact Linear Operators on Normed areas 419

8.4 more Spectral Properties of Compact Linear Operators four28

8.5 Operator Equations Involving Compact Linear Operators 4368.6 Further Theorems of Fredholm Type 442

8.7 Fredholm Alternative 451

**Chapter 9. Spectral Theory of Bounded Self-Adjoint Linear Operators**

9.1 Spectral Properties of finite Self-Adjoint Linear Operators 460

9.2 more Spectral Properties of finite Self-Adjoint Linear Operators 4659.3 Positive Operators 469

9.4 Square Roots of a Positive Operator 476

9.5 Projection Operators 480

9.6 Further Properties of Projections 486

9.7 Spectral Family 492

9.8 Spectral Family of a finite Self-Adjoint operator 497

9.9 Spectral illustration of finite Self-Adjoint Linear Operators 505

9.10 Extension of the Spectral Theorem to Continuous Functions 5129.11 Properties of the Spectral Family of a Bounded Self Adjioint Linear Operator 516

**Chapter 10. Unbounded Linear Operators in Hilbert Space .**

10.1 boundless Linear Operators and their Hilbert-Adjoint Operators 524

10.2Hilbert-Adjoint Operators, rhombohedral and Self-Adjoint Linear Operators 530

10.3 Closed Linear Operators and Closures 53510.4 Spectral Properties of Self-Adjoint Linear Operators 541

10.5 Spectral Representation of Unitary Operators 546

10.6 Spectral Representation of Self-Adjoint Linear Operators 556

10.7 Multiplication Operator and Differentiation Operator 562

**Chapter 11. Unbounded Linear Operators in Quantum Mechanics .**. . . . . 571

11.1 Basic Ideas. States, Observables, Position Operator 572

11.2 Momentum Operator. Heisenberg Uncertainty Principle 576

11.3 Time-Independent Schrodinger Equation 583

11.4 Hamilton Operator 590

11.5 Time-Dependent Schrodinger Equation 598

**Appendix 1. Some Material for Review and Reference**. . . . . . . . . . . . . . 609

A1.1 Sets 609 A1.2 Mappings 613 A1.3 Families 617

A1.4 Equivalence Relations 618 A1.5 Compactness 618

A1.6 Supremum and Infimum 619 A1.7 Cauchy Convergence Criterion 620 A1.8 Groups 622

Appendix 2. Answers to Odd-Numbered Problems. 623

Appendix 3. References. .675

Index . . . . . . . . . . .681

## Post A Comment:

## 0 comments so far,add yours