| Lec. | Date | Section | Topic |
|---|---|---|---|
| 1 | 1/13 | 2.1 | Normed spaces. Banach spaces. Norms on Rn. |
| 2 | 1/15 | 2.1, A.6.2 | Sequence spaces and inequalities. |
| 3 | 1/17 | 2.1, A.6.2 | Sequence spaces. Function spaces. |
| - | 1/20 | - | Martin Luther King Day - no classes |
| 4 | 1/22 | 2.2 | Linear operators. |
| 5 | 1/24 | 2.2, 2.3 | More examples. Finite-dimensional spaces. |
| 6 | 1/27 | 2.4 | Seminorms and Fréchet spaces. |
| 7 | 1/29 | 2.5 | Hahn-Banach extension theorem. |
| 8 | 1/31 | 2.5 2.6 |
Extension theorem for bounded linear functionals. Separation of convex sets. |
| 9 | 2/3 | 2.7 | Dual spaces and weak convergence. |
| 10 | 2/5 | 2.7 3.1 |
Weak-star convergence. Bounded continuous functions. |
| 11 | 2/7 | 3.1 |
Extending a continuous function from a dense set. Pointwise convergence vs. uniform convergence. |
| 12 | 2/10 | 3.2 | Stone-Weierstrass approximation theorem. |
| 13 | 2/12 | 3.3 | Ascoli's compactness theorem. |
| 14 | 2/14 | 3.4 4.1 |
Hölder continuous functions. The uniform boundedness principle. |
| 15 | 2/17 | 4.2 | The open mapping theorem. |
| 16 | 2/19 | 4.3 | Examples. The closed graph theorem. |
| 17 | 2/21 | 4.5 | Compact operators. |
| 18 | 2/24 | Exam 1 covering Chapter 2 and 3. | |
| 19 | 2/26 | 4.4, 4.5 | Adjoint operators. |
| 20 | 2/28 | 5.1 | Hilbert spaces. |
| 21 | 3/2 | 5.2 | Orthogonal projections and orthogonal decomposition. |
| 22 | 3/4 | 5.2 5.3 |
More examples. Continuous linear functionals on Hilbert spaces. |
| 23 | 3/6 | 5.4, 5.5 | Orthonormal sets. |
| 3/8-14 | Spring Break - no classes | ||
| 24 | 3/16 | 5.6 | Positive definite operators. |
| 25 | 3/18 | 5.7 | Weak convergence in Hilbert spaces. |
| 26 | 3/20 | 5.7 6.1 |
An example (5.16) Operators of the form I − compact: a preview. |
| 27 | 3/23 | 6.1 | Fredholm Theory. |
| 28 | 3/25 | 6.1 | Fredholm Theory. |
| 29 | 3/27 | 6.2 | Spectrum of a compact operator. |
| 30 | 3/30 | 6.3 | Self-adjoint and compact self-adjoint operators. |
| 31 | 4/1 | Exam 2 | |
| 32 | 4/2 | 6.3 |
Remarks. Some general properties of the resolvent set and spectrum. |
| 33 | 4/6 | 8.1 | Weak derivatives in R. Examples. |
| 34 | 4/8 | 8.1 | Distributions and weak derivatives. |
| 35 | 4/10 | A.5, 8.2 | Mollifications. |
| 36 | 4/13 | 8.3 | Sobolev Spaces. |
| 37 | 4/15 | 8.3 | Sobolev Spaces. |
| 38 | 4/17 | 8.4-6 | Results on Sobolev spaces. |
| 39 | 4/20 | 8.6 | Embedding theorems. |
| 40 | 4/22 | 8.8 9.1 |
Differentiability properties of Sobolev functions. Elliptic equations. Weak solutions. |
| 41 | 4/24 | 9.1 | Elliptic equations. |
| 42 | 4/27 | 9.1 | Elliptic equations. |
| 43 | 4/29 | Problems. | |
| 44 | 5/1 | Problems. | |