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.

L^{2}.

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.