Here is my current plan for the lectures, it may need to be adjusted a bit as we proceed.

Lecture 1: Feb 5  Outline, metric spaces, normed spaces, Banach spaces, examples.

Lecture 2: Feb 7  Linear maps, boundedness, spaces of bounded linear maps.

Lecture 3: Feb 12  Completion of a normed space. Absolutely summable series.

Lecture 4: Feb 14  Integration, extension of Riemann integral.
(Feb 19 is an MIT Monday)

Lecture 5: Feb 21  Lebesgue integrable functions, integral. Convergence a.e.

Lecture 6: Feb 26  Completeness, monotone convergence, Fatou.

Lecture 7: Feb 28  Dominated convergence, Lebesgue measure etc.

Lecture 8: Mar 5  (pre)Hilbert spaces, CauchySchwarz, examples.

Lecture 9: Mar 7  Bessel's inequality, GramSchidt, orthonormal bases, separability.

Lecture 10: Mar 12  Test 1  on material to Lecture 7.

Lecture 11: Mar 14  Convex sets. Riesz' representation. Projections. Adjoints.

Lecture 12: Mar 19  Compact sets. Weak compactness. Weak convergence.

Lecture 13: Mar 21  Baire's theorem, Uniform Boundedness.

Lecture 14: Apr 2  Completeness of Fourier basis, Fejer kernel

Lecture 15: Apr 4  Open Mapping, Closed Graph theorems.

Lecture 16: Apr 9  Neumann series and invertible operators.

Lecture 17: Apr 11  Compact operators as closure of finite rank ideal.
(Apr 16 is a vacation day)

Lecture 18: Apr 18  Test 2  on material to Lecture 15.

Lecture 19: Apr 23  Fredholm operators, compact perturbations of the identity.

Lecture 20: Apr 25  Spectral theorem for compact selfadjoint operators.

Lecture 21: April 30  Dirichlet problem on an interval

Lecture 22: May 2  More on Dirichlet problem.

Lecture 23: May 7  Harmonic oscillator.

Lecture 24: May 9  Completeness of the Hermit basis.

Lecture 25: May 14  Fourier transform on the line.

Lecture 26: May 16  HahnBanach and review.