Here is my current plan for the lectures, it may need to be adjusted a bit as we proceed.
-
Lecture 1: Feb 3 -- Outline, metric spaces, normed spaces, Banach spaces, examples.
-
Lecture 2: Feb 5 -- Linear maps, boundedness, spaces of bounded linear maps.
-
Lecture 3: Feb 10 -- Lebesgue integrable functions, measure zero.
-
Lecture 4: Feb 12 -- Extension of Riemann integral.
(Feb 17 is an MIT Monday)
-
Lecture 5: Feb 19 -- Completeness, monotone convergence, Fatou.
-
Lecture 6: Feb 24 -- Dominated convergence, Lebesgue measure etc.
-
Lecture 7: Feb 26 -- (pre-)Hilbert spaces, Cauchy-Schwarz, examples.
-
Lecture 8: Mar 3 --
-
Lecture 9: Mar 5 -- Test 1 -- on material to Lecture 7.
-
Lecture 10: Mar 10 -- Bessel's inequality, Gram-Schidt, orthonormal bases, separability.
-
Lecture 11: Mar 12 -- Convex sets. Riesz' representation. Projections. Adjoints.
-
Lecture 12: Mar 17 -- Compact sets. Weak compactness. Weak convergence.
-
Lecture 13: Mar 19 -- Baire's theorem, Uniform Boundedness.
-
Lecture 14: Mar 31 -- Completeness of Fourier basis, Fejer kernel
-
Lecture 15: Apr 2 -- Test 2 -- on material to Lecture 15.
-
Lecture 16: Apr 7 -- Open Mapping, Closed Graph theorems.
-
Lecture 17: Apr 9 -- Neumann series and invertible operators.
-
Lecture 18: Apr 14 -- Compact operators as closure of finite rank ideal.
-
Lecture 19: Apr 16 -- Fredholm operators, compact perturbations of the identity.
(Apr 21 is a vacation)
-
Lecture 20: Apr 23 -- Spectral theorem for compact self-adjoint operators.
-
Lecture 21: Apr 28 -- Dirichlet problem on an interval
-
Lecture 22: Apr 30 -- More on Dirichlet problem.
-
Lecture 23: May 5 -- Fourier transform on the line.
-
Lecture 24: May 7 -- Scattering on the line
-
Lecture 25: May 12 -- Scattering continued
-
Lecture 26: May 14 -- Hahn-Banach and review.