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 4 -- Outline, metric spaces, normed spaces, Banach spaces, examples.
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Lecture 2: Feb 6 -- Linear maps, boundedness, spaces of bounded linear maps.
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Lecture 3: Feb 11 -- Completion of a normed space. Absolutely summable series.
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Lecture 4: Feb 13 -- Integration, extension of Riemann integral.
(Feb 18 is an MIT Monday)
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Lecture 5: Feb 20 -- Lebesgue integrable functions, integral. Convergence a.e.
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Lecture 6: Feb 25 -- Completeness, monotone convergence, Fatou.
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Lecture 7: Feb 27 -- Dominated convergence, Lebesgue measure etc.
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Lecture 8: Mar 4 -- (pre-)Hilbert spaces, Cauchy-Schwarz, examples.
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Lecture 9: Mar 6 -- Test 1 -- on material to Lecture 7.
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Lecture 10: Mar 13 -- Bessel's inequality, Gram-Schidt, orthonormal bases, separability.
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Lecture 11: Mar 18 -- Convex sets. Riesz' representation. Projections. Adjoints.
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Lecture 12: Mar 20 -- 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 1 -- Completeness of Fourier basis, Fejer kernel
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Lecture 15: Apr 3 -- Test 2 -- on material to Lecture 15.
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Lecture 16: Apr 8 -- Open Mapping, Closed Graph theorems.
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Lecture 17: Apr 10 -- Neumann series and invertible operators.
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Lecture 18: Apr 15 -- Compact operators as closure of finite rank ideal.
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Lecture 19: Apr 17 -- Fredholm operators, compact perturbations of the identity.
(Apr 22 is a vacation)
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Lecture 20: Apr 22 -- Spectral theorem for compact self-adjoint operators.
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Lecture 21: Apr 29 -- Dirichlet problem on an interval
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Lecture 22: May 1 -- More on Dirichlet problem.
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Lecture 23: May 6 -- Fourier transform on the line.
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Lecture 24: May 8 -- Scattering on the line
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Lecture 25: May 13 -- Scattering continued
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Lecture 26: May 15 -- Hahn-Banach and review.