Lecture contents for 18.102/18.1021 with suggested reading from the notes.
There are likely to be small modifications as we go along.
  1. Lecture 1: 7 February.
    General outline. Metric spaces, normed and Banach spaces.
    Introduction, Chapter 1, Sections 1-3
    MIT closed on 9 Feb, lecture cancelled.
  2. Lecture 2: 14 February.
    Linear maps, boundedness, spaces of bounded linear maps, brief discussion of Completion of a normed space,
    Chapter 1, Sections 4,5.
  3. Lecture 3: 16 February.
    Lebesgue integrable functions, measure zero. Linearity of Lebesgue space, absolute value.
    Chapter 1, Sections 6,7. Chapter 2, Section 1,2.
  4. Lecture 4: 23 February.
    Lebesgue integral. Completeness.
    Chapter 2, Sections 3,4.
    (Feb 21 is an MIT Monday)
  5. Lecture 5: 28 February.
    Monotone convergence, Fatou. Dominated convergence.
    Chapter 2, Sections 4, 5 and beginning of 6.
  6. Lecture 6: 28 February.
    L2.
    Chapter 2, sections 6,7 and 10.
  7. Lecture 7: 2 March.

    Chapter 3
  8. Lecture 8: 7 March.
    Cauchy-Schwarz, Bessel's inequality, convexity.
    Chapter 3, sections 1 to 8.
    Chapter 3 to Section 9.
  9. Test 1: 9 March.
    On material up to and including Lecture 6
  10. Lecture 9: 14 March.
    Convexity Lemma, Riesz' Representation, adjoints.
    Chapter 3, Sections 8-11.
  11. Lecture 10: 16 March.
    Compact sets. Weak convergence.
    Chapter 3, Section 12.
  12. Lecture 11: 21 March.
    Finite rank and compact operators
    Chapter 3, Sections 14-16
  13. Lecture 12: 23 March.
    Baire's theorem, Uniform Boundedness.
    Chapter 3, Section 15. Chapter 1, Sections 8,9.
  14. Lecture 14: 4 April.
    Neumann series and invertible operators, spectrum of an operator.
    Chapter 3, Sections 15, 16, 17.
  15. Lecture 15: 6 March.
    Spectral theorem for compact self-adjoint operators
    Chapter 3, Section 18.
  16. Lecture 16: 11 April.
    Functional calculus for bounded self-adjoint operators.
    Chapter 3, Sections 17, 18, 19.
  17. Lecture 17: 13 April.
    Polar decomposition, Fredholm operators
    Chapter 3, Sections 21 - 23.
  18. Lecture 18: 20 April.
    Completeness of Fourier basis, Fejér kernel.
    Chapter 4, Sect 1,
  19. Lecture 19: 25 April.
    Test 2 -- on material up to and including Lecture 17
  20. Lecture 20: 27 April.
    The Dirichlet problem on an interval.
    Chapter 4, Sect 2.
  21. Lecture 21: 2 May.
    Fourier transform
    Chapter 4, Sect 7 and 8
  22. Lecture 22: 4 May.
    Fourier inversion
    Chapter 4, Section 14.
  23. Lecture 23: 9 May.
    Convolution and density
    Chapter 4, Section 9.
  24. Lecture 25: 16 May.
    Harmonic oscillator
    Chapter 4, Section 5.
  25. Lecture 24: 11 May.
    Sobolev spaces?
  26. Lecture 26: 18 May.
    Hahn-Banach and review.
    Chapter 1, Section 12.