**Functional Analysis and Summability**

By P.N. Natarajan

**Contents **

Index of Symbols xiii

Preface xvii

1 Some Basic Concepts in Functional Analysis 1

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

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Metric space, examples, Banach space, and examples of

Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Linear Transformations, Linear Functionals, Convexity 29

2.1 Factor spaces, linear transformation, and continuity of a linear

transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Linear functionals, bounded linear functionals, conjugate space 40

2.3 Some results on _nite-dimensional spaces . . . . . . . . . . . 43

2.4 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Hahn{Banach Theorem 55

3.1 Hahn{Banach theorem for the real normed linear space . . . 55

3.2 Hahn{Banach theorem for the complex normed linear space 60

3.3 Some consequences of the Hahn{Banach theorem . . . . . . 63

3.4 Non-uniqueness and uniqueness of the Hahn{Banach extension 70

4 Reexivity 77

4.1 Second conjugate spaces, reexivity . . . . . . . . . . . . . . 77

4.2 Conjugate space of `p, 1 < p < 1 . . . . . . . . . . . . . . . 79

4.3 Conjugate spaces of `1; c0 . . . . . . . . . . . . . . . . . . . . 84

4.4 Conjugate space of C[a; b] . . . . . . . . . . . . . . . . . . . . 88

5 Banach{Steinhaus Theorem 103

5.1 Baire’s theorem: its consequences and applications . . . . . . 103

5.2 Di_erent types of convergence . . . . . . . . . . . . . . . . . 111

5.3 Principle of uniform boundedness (Banach{Steinhaus theorem) 119

5.4 Some consequences and applications of the Banach{Steinhaus

theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6 Closed Graph Theorem and Open Mapping Theorem 133

6.1 Bounded inverse theorem . . . . . . . . . . . . . . . . . . . . 133

6.2 Some consequences of the bounded inverse theorem: Closed

graph theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.3 Interior mapping principle and the open mapping theorem . 145

7 Hilbert Spaces 149

7.1 Hermitian forms, inner product spaces, and Hilbert spaces . 149

7.2 Orthogonality and orthogonal complements . . . . . . . . . . 156

7.3 In_nite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4 Riesz representation theorem . . . . . . . . . . . . . . . . . . 168

8 Silverman{Toeplitz Theorem and Schur’s Theorem 175

8.1 Silverman{Toeplitz theorem . . . . . . . . . . . . . . . . . . 175

8.2 Examples of regular methods . . . . . . . . . . . . . . . . . . 183

8.3 Schur’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.4 Steinhaus theorem and its improvement . . . . . . . . . . . . 191

9 Steinhaus-Type Theorems 199

9.1 Some Steinhaus-type theorems . . . . . . . . . . . . . . . . . 199

9.2 More Steinhaus-type theorems . . . . . . . . . . . . . . . . . 208

Bibliography 215

Index 217

**Preface**

There are excellent books on functional analysis but only a few of them are really introductory. The master work Th_eorie Des Op_erations Lin_eaires (1932) of S. Banach stands out as the _rst and foremost among the several excellent books on functional analysis. Study of divergent series is the foundation of summability theory. Again there are excellent books on summability theory, the outstanding among them being Divergent Series (1949) of G.H. Hardy.

Most of these are terse. My own experience in teaching these topics to several batches of post-graduate students pinpointed to me that they needed a more gentle introduction of these topics. I have written the present book with this point in mind. While writing the book, I have been inuenced by many authors; of whom, special mention has to be made of G. Bachman and L. Narici [1], C. Go_man and G. Pedrick [7], G.H. Hardy [8], I.J. Maddox [13], and G.F. Simmons [24]. Though the book is primarily meant for students of **mathematics** with basic knowledge of real and complex analysis, it will also be useful to students of physics and engineering to get some avour of functional analysis and summability. The topic on hand has many utilities in applied mathematics too. A physicist or engineer, who works on Fourier Series, Fourier transforms, or analytic continuation, can also _nd the book very useful for his/her research.

In this book, spread over nine chapters, a lot of examples have been provided at appropriate places to aid student understanding, and exercises have been given at needed places.

We now briey describe the contents of the various chapters of the book. In Chapter 1, we introduce several basic concepts in functional analysis such as linear space, inner product space, normed linear space, metric space, completeness of a metric space, and Banach space with numerous examples.

Chapter 2 is devoted to a study of factor space, linear transformation, continuity of a linear transformation, linear functionals, bounded linear functionals, and then a conjugate space. Further, we prove some results on _nitedimensional spaces and then study convexity.

In Chapter 3, we prove the Hahn{Banach theorem for real-normed linear spaces and complex-normed linear spaces. Further, we study some consequences of the Hahn{Banach theorem. In Chapter 4, we study second conjugate spaces, reexivity, and proceed to _nd the conjugate spaces of `1, `p, p > 1, c0 and C[a; b]. In Chapter 5, we introduce several types of convergence. We then prove the Banach{Steinhaus theorem (or the uniform boundedness principle). Some consequences and applications of the Banach{Steinhaus theorem are then discussed. In Chapter 6, we study the bounded inverse theorem, closed graph theorem, interior mapping principle, and the open mapping theorem.

In Chapter 7, we introduce Hilbert spaces, orthogonality, orthogonal complements, and orthonormal basis, and establish the Riesz representation theorem for linear functionals on a Hilbert space. With Chapter 7, we conclude the functional analysis component of the present book. With Chapter 8, we commence the summability component of the book. In

Chapter 8, we prove the famous Silverman{Toeplitz theorem on regularity of matrix transformations.We give several examples of regular methods.We then proceed to prove Schur’s theorem. We also record Hahn’s version of Schur’s theorem.

In Chapter 9, we establish the Steinhaus theorem, again a famous result in summability theory. We also prove several Steinhaus-type theorems in the remaining part of the chapter.

In the summability component of the book, we give instances where functional analytic tools like the uniform boundedness principle are used to prove very important theorems.

The author thanks Mr. Boopal Ethirajan for expertly typing the manuscript. Constructive suggestions for the improvement of the book are always welcome.