It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader’s guide stating the needed definitions and basic results in the area and closes with a short description of the problems.

The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most “natural” rather than the most elegant solution is presented.

**Alberto Torchinsky**, Indiana University, Bloomington, IN

Preface

**Part 1. Problems**

Chapter 1. Set Theory and Metric Spaces

Problems

Chapter 2. Measures

Problems

Chapter 3. Lebesgue Measure

Problems

Chapter 4. Measurable and Integrable Functions

Problems

Chapter 5. *L**p *Spaces

Problems

Chapter 6. Sequences of Functions

Problems

Chapter 7. Product Measures

Problems

Chapter 8. Normed Linear Spaces. Functionals

Problems

Chapter 9. Normed Linear Spaces. Linear Operators

Problems

Chapter 10. Hilbert Spaces

Problems

**Part 2. Solutions**

Chapter 11. Set Theory and Metric Spaces

Solutions

Chapter 12. Measures

Solutions

Chapter 13. Lebesgue Measure

Solutions

Chapter 14. Measurable and Integrable Functions

Solutions

Chapter 15. *L**p *Spaces

Solutions

Chapter 16. Sequences of Functions

Solutions

Chapter 17. Product Measures

Solutions

Chapter 18. Normed Linear Spaces. Functionals

Solutions

Chapter 19. Normed Linear Spaces. Linear Operators

Solutions

Chapter 20. Hilbert Spaces

Solutions