# Math 175: Schedule, Syllabus and Homework

Spring quarter 2013

### Course content

#### Week 1: 04/01 - 04/05

• Class 1 (Mon): Organisation, motivation and review of vector spaces (Chapter 1 by Moor Xu's script).
• Class 2 (Wed): Continuing review of vector spaces, Normed spaces (Chapter 2 of course book).
• Class 3 (Fri): Continuing normed spaces. Additional statement: Sphere is compact if and only if the dimension is finite.
• Problem Set 1 (due Fri 4/12), Solution.

#### Week 2: 04/08 - 04/12

• Class 4 (Mon): Differences between closed and complete sets, basic facts on compactness, proof of Theorem 2.13, (Based on Analysis I by Serge Lang and chapter 2.) Inner product spaces (Chapter 1).
• Class 5 (Wed): Motivation inner products: geometric intuition on infinite dimensional vector spaces, outlook how this is used later. Definition inner products. Relation to norms. (Chapter 1)
• Class 6 (Fri): Proof of Cauchy-Schwarz inequality. When does the norm come from an inner product? Parallelogram law. Polarisation identity (Chapter 1). Definition Hilbert und Banach spaces (Chapter 3).
• Problem Set 2 (due Fri 4/19), Solution.

#### Week 3: 04/15 - 04/19

• Class 7 (Mon): The space L^2. Comparing Riemann and Lebesgue measure (partly in the book).
• Class 8 (Wed): Discussion Lebesgue integral. Dominated convergence theorem (page out of Folland Real Analysis). Characterization of null sets (p. 25 in the book). The Cantor set. Naive measure theory fails.
• Class 9 (Fri): Naive measure theory fails part 2. Paradoxon of Bannach-Tarksi. Nearest point property (chapter 3.2 in the book).
• Problem Set 3 (due Fr 4/26) solution

#### Week 4: 04/22 - 04/26

• Class 10 (Mon): Proof of nearest point property. Strong definition of the Sobolev spaces H^1_0 and H^1 (Personal notes).
• Class 11 (Wed): Application of closest point property to minimizing functionals. Associated Euler-Lagrange equation (Personal notes).
• Class 12 (Fri): Euler-Lagrange continued. Multidimensional partial integration (Personal notes). Orthognonal expansions (chapter 4 in the book).
• Problem Set 4 (due Fr 5/3) solution

#### Week 5: 04/29 - 05/03

• Class 13 (Mon): Orthogonal expansions continued. Orthonormal squences and the closest point property. Complete orthonormal sequences (chapter 4 in the book).
• Class 14 (Wed): Repetition: Strong definition of Sobolev space (personal note).
• Class 15 (Fri): Unitary operators and orthogonal complements (chapter 4.3 and 4.3. in the book).
• Takehome Midterm Exam (due Mon 5/6) solution

#### Week 6: 05/06 - 05/10

• Class 16 (Mon): Orthogonal complements. Introduction/ motivation to Fourier series (chapter 5 in the book)
• Class 17 (Wed): Fourier series (personal notes)
• Class 18 (Fri): The representation of the Fejer kernel (chapter 5 in the book). Excursion: differentiation in infinte dimensional normed spaces.
• Problem Set 5 (due Fr, 5/17) solution

#### Week 7: 05/13 - 05/17

• Class 19 (Mon): Dual spaces (chapter 6 in the book).
• Class 20 (Wed): Dual spaces, proof of Riesz theorem .
• Class 21 (Fri): Linear operators (chapter 7 in the book).
• Problem Set 6 (due Fr, 5/24) solution

#### Week 8: 05/20 - 05/24

• Class 22 (Mon): Linear operators. Invertibility. Adjoint. (chapter 7 in the book).
• Class 23 (Wed): Self/adjoint operators. Spectrum (chapter 7 in the book).
• Class 24 (Fri): Compact operators (chapter 8 in the book)
• Problem Set 7 (due Fr, 5/31) solution

#### Week 9: 05/27 - 05/31

• Class 25 (Wed): Compact operators. Spectral theorem for compact self-adjoint operators (chapter 8 in the book)
• Class 26 (Fri): Proof of Spectral theorem for compact self-adjoint operators (chapter 8 in the book)
• Problem Set 8 solution

#### Week 10: 06/03 - 06/07

• Class 27 (Mon): Application of spectral theorem to partiell differential equations. Stationary equations. Laplace equation. Derivation of the solution operator: Integraloperator with the Greens function k(x,t). Integraloperator is Hilbert-Schmitt, hence compact. The regular Sturm Liouville operator (Chapter 10, page 119 and 120, personal note)
• Class 28 (Wed): Greens function for the regular Sturm Liouville operator (Combination of Theorem 10.3 and 10.5 in the book). Formal soltution of time dependent equations using functional/operator calculus (personal note).