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

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