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