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 Serious mathematics, written with the reader in mind.
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Functional Analysis Volume I: A Gentle Introduction

by Dzung Minh Ha


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640 pages, Hardcover

Table of Contents

Chapter 0: three important inequalities 1

Chapter 1: Metric and topological spaces 6

1.1 Metrics and metric spaces
1.2 Open and closed sets
1.3 Topological spaces
1.4 Continuous functions
1.5 Open sets and continuity
1.6 Some important topological concepts
1.7 Convergence of sequences in metric spaces
1.8 Completeness
1.9 Density, separability, and approximation
1.10 Metric space completions
1.11 Compactness
1.12 The Banach fixed point theorem
1.13 Baire's category theorem

Chapter 2: Normed spaces 116

2.1 Linear operators on function spaces
2.2 Hamel bases
2.3 Norms and normed spaces
2.4 Topological concepts in normed spaces
2.5 Topological vector spaces
2.6 Kolmogorov's theorem
2.7 Banach spaces
2.8 Infinite series in normed spaces
2.9 Schauder bases
2.10 Linear functionals and hyperplanes
2.11 Constructing new normed spaces

Chapter 3: Operators on normed spaces 206

3.1 Continuous linear maps
3.2 Integral operators
3.3 Linear homeomorphisms
3.4 Three important theorems
3.5 The normed space B(X,Y)
3.6 Complementary subspaces and projections
3.7 Riesz's lemma
3.8 The spectrum of a bounded linear operator
3.9 Continuous linear functionals and dual spaces

Chapter 4: Inner product spaces 291

4.1 Definitions and examples
4.2 Orthogonality
4.3 Unitary isomorphisms
4.4 Inner product spaces: three problems
4.5 Three characterizations for Hilbert spaces
4.6 Hilbert bases

Chapter 5: The Banach space C(X) 365

5.1 The Arzela-Ascoli theorem
5.2 Korovkin's theorem and the Weierstrass approximation theorem
5.3 Sub-algebras
5.4 The Stone-Weierstrass theorem

Chapter 6: Additional topics 415

6.1 The Baire-Osgood theorem
6.2 Gram determinants and Muntz's theorem
6.3 Differential equations

Appendix A: Set theory and functions 464

A.1 Sets
A.2 Relations
A.3 Zorn's lemma and the axiom of choice
A.4 Functions
A.5 Cardinality
A.6 The axiom of completeness on R

Appendix B: Mostly linear algebra (a brief review) 480

B.1 Polynomials and sequences
B.2 Vector spaces
B.3 Linear independence and span
B.4 Bases and dimension
B.5 Linear transformations
B.6 Partial derivatives and the mean value theorem
B.7 Riemann integrals

Appendix C: Some technical results 491

Appendix D: Solutions to odd-numbered exercises 494

Bibliography 618

Notation 622

Index 624

ISBN  978-0-9715766-1-2

640 pages, Hardcover, smythe-sewn binding

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April 20, 2006