Introduction to Functional Analysis - Second Edition
(C. R. de Oliveira; February/2008)
Contents
(about 215 pages)1. Normed Spaces
2. Compactness and Completion
2.1 Compactness and Dimension
2.2 Completion of Normed Spaces
3. Linear Operators
3.1 Separable Spaces
3.2 Linear Operators
4. Bounded Operators and Dual Space
5. Banach Fixed Point
6. Baire Category Theorem
7. Uniform Boundedness Principle
8. Open Mapping Theorem
9. Closed Graph Theorem
10. Hahn-Banach Theorem
10.1 Zorn Lemma
10.2 Hahn-Banach
11. Proof of Hahn-Banach
12. Applications of Hahn-Banach Theorem
13. Adjoint Operators in Normed Spaces
14. Weak Convergence
15. Weak Topologies
15.1 Weak Topologies
15.2 Alaoglu Theorem
16. Reflexive Spaces and Sequential Compactness
17. Hilbert Spaces
17.1 Inner Product
17.2 Orthogonality
18. Orthogonal Projection
18.1 Parallelogram Rule
18.2 Orthogonal Projection
19. Riesz Representation in Hilbert Spaces
19.1 Riesz Representation
19.2 Hilbert Adjoint and Lax-Milgram
20. Self-Adjoint Operators
21. Orthonormal Bases
22. Fourier Series
22.1 Fourier Series
22.2 Integration in Hilbert Spaces
23. Operations in Banach Spaces
23.1 Direct Sum
23.2 Quotient Space
24. Compact Operators
25. Compact Operators in Hilbert Spaces
26. Hilbert-Schmidt Operators
27. Spectrum
28. Spectral Classification
29. Spectrum of Self-Adjoint Operators
30. Spectrum of Compact Operators
30.1 Compact Operators
30.2 Normal Compact Operators