Contents

PREFACE TO THE REVISED EDITION xv

FROM THE PREFACE TO THE FIRST EDITION xix

Chapter 1. BASIC NOTIONS 1

§1.1. A problem from diﬀerential equations 1

§1.2. An examination of the results 6

§1.3. Examples of Banach spaces 9

§1.4. Fourier series 17

§1.5. Problems 24

Chapter 2. DUALITY 29

§2.1. The Riesz representation theorem 29

§2.2. The Hahn-Banach theorem 33

§2.3. Consequences of the Hahn-Banach theorem 36

§2.4. Examples of dual spaces 39

§2.5. Problems 51

Chapter 3. LINEAR OPERATORS 55

§3.1. Basic properties 55

§3.2. The adjoint operator 57

§3.3. Annihilators 59

§3.4. The inverse operator 60

§3.5. Operators with closed ranges 66

§3.6. The uniform boundedness principle 71

ix