Lecture Notes

Some lecture notes include homework assignments plus solutions.

1 Linear spaces, metric spaces, normed spaces (PDF)
2 Linear maps between normed spaces (PDF)
3 Banach spaces (PDF)
4 Lebesgue integrability (PDF)
5 Lebesgue integrable functions form a linear space (PDF)
6 Null functions (PDF)
7 Monotonicity, Fatou's Lemma and Lebesgue dominated convergence (PDF)
8 Hilbert spaces (PDF)
9 Baire's theorem and an application (PDF)
10 Bessel's inequality (PDF)
11 Closed convex sets and minimizing length (PDF)
12 Compact sets. Weak convergence. Weak compactness (PDF)
13 Baire's theorem. Uniform boundedness. Boundedness of weakly convergent sequences (PDF)
14 Fourier series and L2 (PDF)
15 Open mapping and closed graph theorems (PDF)
16 Bounded operators. Unitary operators. Finite rank operators (PDF)
17 The second test (PDF)
18 Compact operators (PDF)
19 Fredholm operators (PDF)
20 Completeness of the eigenfunctions (PDF)
21 Dirichlet problem for a real potential on an interval (PDF)
22 Dirichlet problem (cont.) (PDF)
23 Harmonic oscillator (PDF)
24 Completeness of Hermite basis (PDF)
25 The fourier transform on the line (PDF)
26 Hahn-Banach and review (PDF)