Functional analysis

Contents......Page 2
Comments and course information......Page 5
Part 1.  Hahn-Banach Theorem and Applications......Page 7
Lecture 1. Linear spaces and the Hahn Banach Theorem......Page 9
Lecture 2. Geometric Hahn-Banach Theorems......Page 13
Lecture 3. Applications of Hahn-Banach......Page 17
Part 2.  Banach Spaces......Page 21
Lecture 4. Normed and Banach Spaces......Page 23
Lecture 5. Noncompactness of the Ball and Uniform Convexity......Page 27
Lecture 6. Linear Functionals on a Banach Space......Page 31
Homework I......Page 35
Part 3.  Hilbert Spaces and Applications......Page 40
Lecture 8. Scalar Products and Hilbert Spaces......Page 42
Lecture 9. Riesz-Frechet and Lax-Milgram Theorems......Page 46
Lecture 10. Geometry of a Hilbert space and Gram-Schmidt process......Page 50
Part 4.  Locally Convex Spaces......Page 54
Lecture 11. Locally Convex Spaces and Spaces of Test Functions......Page 56
Lecture 12. Generation of a LCS by seminorms and Fréchet Spaces......Page 58
Lecture 13. The dual of an LCS......Page 62
Lecture 14. Spaces of distributions......Page 66
Lecture 15. Applications: solving some PDE\'s......Page 70
Lecture 16. The Dirichlet problem......Page 72
Part 5.  Weak Convergence and Weak Topology......Page 76
Lecture 17. Dual of a Banach space......Page 78
Lecture 18. Riesz-Kakutani theorem......Page 82
Lecture 19. Weak convergence......Page 86
Lecture 20. Weak sequential compactness, weak* convergence and the weak topology......Page 90
Lecture 21. An application: positive harmonic functions......Page 94
Presentation topics......Page 96
Homework II......Page 99
Part 6.  Convexity......Page 104
Lecture 22. Convex sets in a Banach space......Page 106
Lecture 23. Convex sets in a Banach space (II)......Page 110
Lecture 24. Krein-Milman and Stone-Weierstrass......Page 114
Lecture 25. Choquet type theorems......Page 118
Part 7.  Bounded Linear Maps......Page 122
Lecture 26. Bounded Linear Maps......Page 124
Lecture 27. Principle of Uniform Boundedness and Open Mapping Theorem......Page 128
Lecture 28. The Spectrum of a Linear Map......Page 132
Lecture 29. Some examples......Page 136
Part 8.  Compact Linear Maps......Page 140
Lecture 30. Compact Maps......Page 142
Lecture 31. Fredholm alternative......Page 146
Homework III......Page 150
Part 9.  Compact Linear Maps in Hilbert Space......Page 157
Lecture 33. Compact Symmetric Operators......Page 159
Lecture 34. Min-Max......Page 163
Lecture 35. Functional calculus and polar decomposition......Page 167