Lecture contents for 18.102/18.1021 with suggested reading from the notes.
There are likely to be small modifications as we go along.
- 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.
- 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.
- 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.
- Lecture 4: 23 February.
Lebesgue integral. Completeness.
Chapter 2, Sections 3,4.
(Feb 21 is an MIT Monday)
- Lecture 5: 28 February.
Monotone convergence, Fatou. Dominated convergence.
Chapter 2, Sections 4, 5 and beginning of 6.
- Lecture 6: 28 February.
L2.
Chapter 2, sections 6,7 and 10.
- Lecture 7: 2 March.
Chapter 3
- Lecture 8: 7 March.
Cauchy-Schwarz, Bessel's inequality, convexity.
Chapter 3, sections 1 to 8.
Chapter 3 to Section 9.
- Test 1: 9 March.
On material up to and including Lecture 6
- Lecture 9: 14 March.
Convexity Lemma, Riesz' Representation, adjoints.
Chapter 3, Sections 8-11.
- Lecture 10: 16 March.
Compact sets. Weak convergence.
Chapter 3, Section 12.
- Lecture 11: 21 March.
Finite rank and compact operators
Chapter 3, Sections 14-16
- Lecture 12: 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.
Fourier inversion
Chapter 4, Section 14.
- Lecture 23: 9 May.
Convolution and density
Chapter 4, Section 9.
- Lecture 25: 16 May.
Harmonic oscillator
Chapter 4, Section 5.
- Lecture 24: 11 May.
Sobolev spaces?
- Lecture 26: 18 May.
Hahn-Banach and review.
Chapter 1, Section 12.