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Functional Analysis—An Elementary Introduction
Markus Haase
Price
1160.00
ISBN
9781470454739
Language
English
Pages
392
Format
Paperback
Dimensions
180 x 240 mm
Year of Publishing
2020
Territorial Rights
Restricted
Imprint
Universities Press
Catalogues

This book introduces functional analysis at an elementary level without assuming any background in real analysis, for example on metric spaces or Lebesgue integration. It focuses on concepts and methods relevant in applied contexts such as variational methods on Hilbert spaces, Neumann series, eigenvalue expansions for compact self-adjoint operators, weak differentiation and Sobolev spaces on intervals, and model applications to differential and integral equations. Beyond that, the final chapters on the uniform boundedness theorem, the open mapping theorem and the Hahn–Banach theorem provide a stepping-stone to more advanced texts.

The exposition is clear and rigorous, featuring full and detailed proofs. Many examples illustrate the new notions and results. Each chapter concludes with a large collection of exercises, some of which are referred to in the margin of the text, tailor-made in order to guide the student digesting the new material. Optional sections and chapters supplement the mandatory parts and allow for modular teaching spanning from basic to honors track level.

Markus Haase, Delft University of Technology, Delft, The Netherlands
• Preface    14
• Chapter 1. Inner product spaces    20
• Chapter 2. Normed spaces    34
• Chapter 3. Distance and approximation    56
• Chapter 4. Continuity and compactness    74
• Chapter 5. Banach spaces    98
• Chapter 6. The contraction principle    112
• Chapter 7. The Lebesgue spaces    126
• Chapter 8. Hilbert space fundamentals    148
• Chapter 9. Approximation theory and Fourier analysis    166
• Chapter 10. Sobolev spaces and the Poisson problem    196
• Chapter 11. Operator theory I    212
• Chapter 12. Operator theory II    230
• Chapter 13. Spectral theory of compact self-adjoint operators    250
• Chapter 14. Applications of the spectral theorem    266
• Chapter 15. Baire’s theorem and its consequences    280
• Chapter 16. Duality and the Hahn-Banach theorem    296
• Historical remarks    324
• Appendix A. Background    330
• Appendix B. The completion of a metric space    352
• Appendix C. Bernstein’s proof of Weierstrass’ theorem    358
• Appendix D. Smooth cutoff functions    362
• Appendix E. Some topics from Fourier analysis    364
• Appendix F. General orthonormal systems    370
• Bibliography    374
• Symbol Index    378
• Subject Index    380
• Author Index    390
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