Course syllabus
Lectures
(This is a tentative list, which will be continuously updated as the course progresses.)
| Lecture |
Content |
Sections [EW] | Important Theorems |
|---|---|---|---|
| 44.1 |
Introduction/Motivation. Normed vector spaces. |
2.1.1 |
Continuity of norm; Equivalence of norms; On finite-dimensional space, all norms are equivalent. |
| 44.2 |
Semi-norms and quotient norms Banach spaces |
2.1.2 2.2 |
Criteria for quotient semi-norms being a norm. Concrete computation of quotient norms. |
| 44.3 |
Subspaces and quotients of Banach space Examples: B(X), Cb(X), C0(X) |
2.2.1 |
Quotient of Banach space is Banach space. |
| 45.1 |
Completion of normed spaces. Non-compactness of unit ball in infinite-dimensional Banach spaces Arzela-Ascoli |
2.2.2 2.2.3 2.3.1 |
|
| 45.2 |
Stone-Weiterstraß Bounded operators and linear functionals. |
2.3.2 2.4 |
Compactness. Examples of compact and non-compact subsets in Banach spaces. |
| 45.3 |
Hilbert spaces |
3.1.1 |
|
| 46.1 |
Examples of Hilbert spaces, Riesz-Fischer theorem, Unitary Operators, Uniform Convexity, Hilbert spaces are uniformly convex |
3.1.1 (ct'd) 3.1.2 |
|
| 46.2 | Closest point in closed, convex set in uniformly convex Banach space; orthogonal complement |
3.1.2 (ct'd) |
|
| 46.3 | Baire Category Theorem |
4.1.1 4.2.1
|
|
| 47.1 |
G-delta and F-sigma sets, Open Mapping Theorem, Existence of Bounded Inverse. |
4.2.2 4.2.3 |
|
| 47.2 |
Banach-Steinhaus (Uniform Boundedness Principle), strong convergence, Closed Graph Theorem |
4.1 4.2.3 |
|
| 47.3 |
Hahn-Banach Theorem and consequences |
7.1.1 7.1.2
|
|
| 48.1 |
Isometric embedding into the bidual, reflexivity |
7.1.3 |
|
| 48.2 |
Banach Limit. Amenable Groups, Banach-Tarski Paradox |
7.2 |
|
| 48.3 |
Weak/Weak* topology |
8.1 |
|
| 49.1 | Banach-Alaoglu (weak* compactness of unit ball in dual space), weak/weak* topology are Hausdorff, weak* continuous functionals |
8.1.1 8.1.2 |
|
| 49.2 | Dual spaces of Lp-spaces, pairing between Lp-spaces |
7.3 |
|
| 49.3 | Riesz theorem for C(X) |
|
|
| 50.1 |
|
||
| 50 | The space C(X), its dual space and application to equip-distributions for points in X. |
2.3.1-2.3.3. 8.2.
|
Continuation of weak*-compactness and application to sequences of measures. |
| W2/W3, 2024 |
Oral examination
|
Recommended exercises
There are so called Essential Exercises in the book - please try to work them out. Exercises marked with * might be a bit difficult. Some extra exercises together with assignments exercises will be given - see Canvas/Assignments.