Contents

Preface xi

Part 0. Preliminaries

Chapter 0. A first look at Banach and Hilbert spaces 3

§0.1. Warm up: Metric and topological spaces 3

§0.2. The Banach space of continuous functions 14

§0.3. The geometry of Hilbert spaces 21

§0.4. Completeness 26

§0.5. Bounded operators 27

§0.6. Lebesgue

Lp

spaces 30

§0.7. Appendix: The uniform boundedness principle 38

Part 1. Mathematical Foundations of Quantum Mechanics

Chapter 1. Hilbert spaces 43

§1.1. Hilbert spaces 43

§1.2. Orthonormal bases 45

§1.3. The projection theorem and the Riesz lemma 49

§1.4. Orthogonal sums and tensor products 52

§1.5. The

C∗

algebra of bounded linear operators 54

§1.6. Weak and strong convergence 55

§1.7. Appendix: The Stone–Weierstraß theorem 59

Chapter 2. Self-adjointness and spectrum 63

vii