## Table of Contents

**Table of Contents**

Preface

1. Non-Archimedean Valued Fields

1.1 Introduction

1.2 Non-Archimedean Valued Fields

1.2.1 Non-Archimedean Valued Fields

1.2.2 The t-Vector Space

1.3 Construction of Qp

1.3.1 Introduction

1.3.2 The Field Qp

1.3.3 Convergence of Power Series over Qp

1.4 Construction of K ((x))

1.5 Bibliographical Notes

2. Non-Archimedean Banach and Hilbert Spaces

2.1 Non-Archimedean Banach Spaces

2.1.1 Basic Definitions

2.1.2 Examples of Non-Archimedean Banach Spaces

2.2 Free Banach Spaces

2.2.1 Definitions

2.2.2 Examples

2.3 Non-Archimedean Hilbert Spaces

2.3.1 Introduction

2.3.2 Non-Archimedean Hilbert Spaces

2.3.3 The Hilbert Space Ew1 x Ew2 x… x Ew1

2.4 Bibliographical Notes

3. Non-Archimedean Bounded Linear Operators

3.1 Introduction

3.2 Bounded Linear Operators on Non-Archimedean Banach Spaces

3.2.1 Basic Definitions

3.2.2 Examples

3.2.3 The Banach Algebra B (X)

3.2.4 Further Properties of Bounded Linear Operators

3.3 Bounded Linear Operators on Non-Archimedean Hilbert Spaces

3.3.1 Introduction

3.3.2 Representation of Bounded Operators By Infinite Matrices

3.3.3 Existence of the Adjoint

3.3.4 Examples of Bounded Operators with no Adjoint

3.4 Perturbation of Bases

3.4.1 Example

3.5 Hilbert-Schmidt Operators

3.5.1 Basic Definitions

3.5.2 Further Properties of Hilbert-Schmidt Operators

3.5.3 Completely Continuous Operators

3.5.4 Trace

3.5.5 Examples

3.6 Open Problems

3.7 Bibliographical Notes

4. Non-Archimedean Unbounded Linear Operators

4.1 Introduction

4.2 Basic Definitions

4.2.1 Example

4.2.2 Existence of the Adjoint

4.2.3 Examples of Unbounded Operators With no Adjoint

4.3 Closed Linear Operators on Ew

4.4 Diagonal Operators on Ew

4.5 Open Problems

4.6 Bibliographical Notes

5. Non-Archimedean Bilinear Forms

5.1 Introduction

5.2 Basic Definitions

5.2.1 Continuous Linear Functionals on Ew

5.2.2 Bounded Bilinear Forms on Ew x Ew

5.2.3 Unbounded Bilinear Forms on Ew x Ew

5.3 Closed and Closable non-Archimedean Bilinear Forms

5.3.1 Closedness of the Form Sum

5.3.2 Construction of a non-Archimedean Hilbert Space Using Quadratic Forms

5.3.3 Further Properties of the Closure

5.4 Representation of Bilinear Forms on Ew x Ew by Linear Operators

5.5 Bibliographical Notes

6. Functions of Linear Operators on Ew

6.1 Introduction

6.2 Products and Sums of Diagonal Operators

6.3 Integer Powers of Diagonal Operators

6.4 Functions of Self-Adjoint Operators

6.5 Functions of Some Symmetric Square Matrices Over Qp x Qp

6.5.1 The Powers of the Matrix T

6.5.2 Exponential of the Matrix T

6.6 Open Problems

6.7 Bibliographical Notes

7. One-Parameter Family of Bounded Linear Operators on Free Banach Spaces

7.1 Introduction

7.2 Basic Definitions

7.3 Properties of Non-Archimedean Co-Groups

7.4 Existence of Solutions to Some p-adic Differential Equations

7.5 Open Problems

7.6 Bibliographical Notes

References

Index