Functional analysis is primarily concerned with infinite-dimensional linear(vector) spaces, mainly function spaces whose "points" are functions, andmappings between them, usually called operators or, functionals if the rangeis on the real line or in the complex plane. It was invented and developedin the last years of the nineteenth century and the first few decades of thetwentieth century. During the early period of its development, the originalpurpose of functional analysis was to use a framework which allows the studyof differential and integral equations to be considered in the same formulation(cf. [6]). Later, functional analysis developed rapidly as in-depth study and in-terconnection on spectral theory of ordinary and partial differential equations,potential theory, Fourier expansions, and applied mathematical techniques,especially, on the influence of mathematical physics and quantum mechan-ics.

Preface

List of Symbols

Chapter 1 Metric Spaces

1.1 Preliminaries

1.2 Definitions and Examples

1.3 Convergence in a Metric Space

1.4 Sets in a Metric Space

1.5 Complete Metric Spaces

1.6 Continuous Mappings on Metric Spaces

1.7 Compact Metric Spaces

1.8 The Contraction Mapping Principle

Chapter 2 Normed Linear Spaces. Banach Spaces

2.1 Review of Linear Spaces

2.2 Norm in a Linear Space

2.3 Examples of Normed Linear Spaces

2.4 Finite Dimensional Normed Linear Spaces

2.5 Linear Subspaces of Normed Linear Spaces

2.6 Quotient Spaces

2.7 The Weierstrass Approximation Theorem

Chapter 3 Inner Product Spaces. Hilbert Spaces

3.1 Inner Products

3.2 Orthogonality

3.3 Orthonormal Systems

3.4 Fourier Series

Chapter 4 Linear Operators. Fundamental Theorems

4.1 Continuous Linear Operators and Functionals

4.2 Spaces of Bounded Linear Operators and Dual Spaces 

4.3 The Banach-Steinhaus Theorem

4.4 Inverses of Operators. The Banach Theorem

4.5 The Hahn-Banach Theorem

4.6 Strong and Weak Convergence

Chapter 5 Linear Operators on Hilbert Spaces

5.1 Adjoint Operators. The Lax-Milgram Theorem

5.2 Spectral Theorem for Self-adjoint Compact Operators

Bibliography

Index