Lecture contents for 18.102/18.1021 with suggested reading from the notes.
There may be small modifications as we go along.
- Lecture 1: 7 February.
General outline. Metric spaces, normed spaces, Banach spaces, examples.
Introduction, Chapter 1, Sections 1-3
- Lecture 2: 9 February.
Linear maps, boundedness, spaces of bounded linear maps.
Chapter 1, Sections 4,5.
- Lecture 3: 14 February.
Completion of a normed space, Lebesgue integrable functions, measure zero.
Chapter 1, Sections 6,7. Chapter 2, Section 1.
- Lecture 4: 16 February.
Linearity of Lebesgue space, absolute value. Monotonic sequence, Lebesgue integral.
Chapter 2, Sections 2,3.
(Feb 21 is an MIT Monday)
- Lecture 5: 23 February.
Completeness, monotone convergence, Fatou.
Chapter 2, Sections 4, 5 and beginning of 6.
- Lecture 6: 28 February.
Fatou, Dominated convergence, Lebesgue measure, L2.
Chapter 2, sections 6,7 and 10.
- Lecture 7: 2 March.
(pre-)Hilbert spaces, Cauchy-Schwarz, Bessel's inequality.
Chapter 3 sections 1 to 4.
- Lecture 8: 7 March.
Convexity and projections
Chapter 3 to Section 9.
- Lecture 9: 9 March.
Test 1 -- on material up to and including Lecture 6
- Lecture 10: 14 March.
Riesz' Representation, adjoints.
Chapter 3, Sections 10,11.
- Lecture 11: 16 March.
Compact sets. Weak convergence.
Chapter 3, Section 12.
- Lecture 12: 21 March.
Finite rank and compact operators
Chapter 3, Sections 14-16
- Lecture 13: 23 March.
Baire's theorem, Uniform Boundedness.
Chapter 3, Section 15. Chapter 1, Sections 8,9.
- Lecture 14: 4 April.
Neumann series and invertible operators, spectrum of an operator.
Chapter 3, Sections 15, 16, 17.
- Lecture 15: 6 March.
Spectral theorem for compact self-adjoint operators
Chapter 3, Section 18.
- Lecture 16: 11 April.
Functional calculus for bounded self-adjoint operators.
Chapter 3, Sections 17, 18, 19.
- Lecture 17: 13 April.
Polar decomposition, Fredholm operators
Chapter 3, Sections 21 - 23.
- Lecture 18: 20 April.
Completeness of Fourier basis, Fejér kernel.
Chapter 4, Sect 1,
- Lecture 19: 25 April.
Test 2 -- on material up to and including Lecture 17
- Lecture 20: 27 April.
The Dirichlet problem on an interval.
Chapter 4, Sect 2.
- Lecture 21: 2 May.
Fourier transform -- Chapter 4, Sect 7 (and 8).
- Lecture 22: 4 May.
Open mapping and closed graph theorems.
Chapter 1, Sects 10 and 11.
- Lecture 23: 9 May.
Convolution and density
Chapter 4, Section 9.
- Lecture 24: 11 May.
Fourier inversion
Chapter 4, Section 14.
- Lecture 25: 16 May.
Harmonic oscillator
Chapter 4, Section 5.
- Lecture 26: 18 May.
Hahn-Banach and review.
Chapter 1, Section 12.