Exercises in Functional Analysis

1. Front Matter

   Pages i-x

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2. Normed spaces

   1. Front Matter

      Pages 1-1

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   2. Open, closed, and bounded sets in normed spaces

      • Constantin Costara, Dumitru Popa

      Pages 3-35

   3. Linear and continuous operators on normed spaces

      • Constantin Costara, Dumitru Popa

      Pages 36-67

   4. Linear and continuous functionals. Reflexive spaces

      • Constantin Costara, Dumitru Popa

      Pages 68-85

   5. The distance between sets in Banach spaces

      • Constantin Costara, Dumitru Popa

      Pages 86-106

   6. Compactness in Banach spaces. Compact operators

      • Constantin Costara, Dumitru Popa

      Pages 107-146

   7. The Uniform Boundedness Principle

      • Constantin Costara, Dumitru Popa

      Pages 147-174

   8. The Hahn—Banach theorem

      • Constantin Costara, Dumitru Popa

      Pages 175-194

   9. Applications for the Hahn—Banach theorem

      • Constantin Costara, Dumitru Popa

      Pages 195-212

   10. Baire’s category. The open mapping and closed graph theorems

       • Constantin Costara, Dumitru Popa

       Pages 213-240

3. Hilbert spaces

   1. Front Matter

      Pages 241-241

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   2. Hilbert spaces, general theory

      • Constantin Costara, Dumitru Popa

      Pages 243-270

   3. The projection in Hilbert spaces

      • Constantin Costara, Dumitru Popa

      Pages 271-304

   4. Linear and continuous operators on Hilbert spaces

      • Constantin Costara, Dumitru Popa

      Pages 305-365

4. General topological spaces

   1. Front Matter

      Pages 366-366

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   2. Linear topological and locally convex spaces

      • Constantin Costara, Dumitru Popa

      Pages 368-402

   3. The weak topologies

      • Constantin Costara, Dumitru Popa

      Pages 403-443

5. Back Matter

   Pages 444-454

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The understanding of results and notions for a student in mathematics requires solving ex­ ercises. The exercises are also meant to test the reader's understanding of the text material, and to enhance the skill in doing calculations. This book is written with these three things in mind. It is a collection of more than 450 exercises in Functional Analysis, meant to help a student understand much better the basic facts which are usually presented in an introductory course in Functional Analysis. Another goal of this book is to help the reader to understand the richness of ideas and techniques which Functional Analysis offers, by providing various exercises, from different topics, from simple ones to, perhaps, more difficult ones. We also hope that some of the exercises herein can be of some help to the teacher of Functional Analysis as seminar tools, and to anyone who is interested in seeing some applications of Functional Analysis. To what extent we have managed to achieve these goals is for the reader to decide.

• Hilbert space
• banach spaces
• boundary element method
• compactness
• function
• functional
• functional analysis
• graph
• locally convex space
• mathematics
• presentation
• sets
• techniques
• theorem
• tool

From the reviews:

"The present book is one of the few books available with problems in functional analysis. … the authors have done a good job in crediting the original sources of the problems. … The layout and the exposition of the book is very good. The book will be of great help to graduate students in mathematics … . will also be very helpful to research mathematicians in analysis … . I strongly recommend the book to any person who is interested in analysis and its applications." (C. D. Aliprantis, Mathematical Reviews, Issue 2005 b)

• Department of Mathematics, Ovidius University of Constanta, Constanta, Romania

  Constantin Costara, Dumitru Popa

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