Functional Analysis and Summability

P.N. Natarajan

• Publisher: Chapman and Hall/CRC
• Year: 2020
• ISBN:
  9781000191493
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

There are excellent books on both functional analysis and summability. Most of them are very terse. In Functional Analysis and Summability, the author makes a sincere attempt for a gentle introduction of these topics to students. In the functional analysis component of the book, the Hahn–Banach theorem, Banach–Steinhaus theorem (or uniform boundedness principle), the open mapping theorem, the closed graph theorem, and the Riesz representation theorem are highlighted. In the summability component of the book, the Silverman–Toeplitz theorem, Schur's theorem, the Steinhaus theorem, and the Steinhaus-type theorems are proved. The utility of functional analytic tools like the uniform boundedness principle to prove some results in summability theory is also pointed out.

Features

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• A gentle introduction of the topics to the students is attempted.

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• Basic results of functional analysis and summability theory and their applications are highlighted.

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• Many examples are provided in the text.

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• Each chapter ends with useful exercises.

This book will be useful to postgraduate students, pre-research level students, and research scholars in mathematics. Students of physics and engineering will also find this book useful since topics in the book also have applications in related areas.

Chapter 1

Some Basic Concepts in Functional Analysis

1.1Linear space, inner product space, and normed linear space: Examples

We now recall a few definitions and concepts from linear algebra that are needed in the sequel.

Definition 1. A linear space or vector space over a field K is a quadruple (X, K, +, ·), where X is a non-empty set, + is a mapping (x, y) → x + y of X × X → X, · is a mapping (α, x) → α · x of K × X → X, and the following conditions are fulfilled:

(i)x + (y + z) = (x + y) + z, for all x, y, z ∈ X;

(ii)x + y = y + x, for all x, y ∈ X;

(iii)there exists θ ∈ X such that x + θ = x, for all x ∈ X;

(iv)for each x ∈ X, there exists an element denoted by −x ∈ X such that...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• About the Author
• Index of Symbols
• Preface
• 1. Some Basic Concepts in Functional Analysis
• 2. Linear Transformations, Linear Functionals, Convexity
• 3. Hahn–Banach Theorem
• 4. Reflexivity
• 5. Banach–Steinhaus Theorem
• 6. Closed Graph Theorem and Open Mapping Theorem
• 7. Hilbert Spaces
• 8. Silverman–Toeplitz Theorem and Schur's Theorem
• 9. Steinhaus-Type Theorems
• Bibliography
• Index