Functional Analysis

This book is written to serve as a senior or beginning graduate text. Although the student should not expect to learn the value of a subject at the time that he is learning the subject itself, we have tried to give, in problems and examples, some of the applications of functional analysis. To mention only two, a proof is given of a version of the Riemann mapping theorem, as is a proof of the existence of a continuous function whose Fourier series diverges. 1. INTRODUCTION 2. LINEAR SPACE 3. SEPARATION 4. PARANORMS AND SEMINORMS 5. SEMIMETRIC SPACE 6. LINEAR SEMIMETRIC SPACE 7. BANACH SPACE 8. HILBERT SPACE 9. TOPOLOGY AND NETS 10. LINEAR TOPOLOGY 11. THE CLOSED GRAPH THEOREM 12. LOCAL CONVEXITY 13. DUALITY 14. BANACH ALGEBRA