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