We recommend that you read the lecture
concurrently with (or prior to) the lectures. If you only read these
sources occasionally and after the fact (e.g. when your homework is due)
then you will not get the most out of the course.
- Week 1: Metric spaces, convergence of sequences, open and closed sets, relatively open and closed sets
- Week 2: Cauchy sequences, complete metric spaces, compact sets, Heine-Borel theorem, continuity, connected sets.
- Week 3: Pointwise and uniform convergence of sequences and series of functions. Uniform convergence, continuity, and integration; Weierstrass M-test.
- Weeks 4/5: Uniform convergence and derivatives; Weierstrass approximation theorem; power series; exponential and log functions; trig functions.
- Week 6: Periodic functions, trigonometric polynomials, trigonometric Weierstrass approximation theorem, Fourier series, Plancherel theorem
- Week 7: Review of linear transformations, differentiation in several variable calculus, Clairaut's theorem, chain rule, inverse function theorem, implicit function theorem
- Weeks 8/9: Outer measure; measurable sets; Lebesgue measure; measurable functions; simple functions
- Week 10: The Lebesgue integral; monotone convergence theorem, Fatou's lemma, and dominated convergence theorem; comparison with Riemann integral; Fubini's theorem
We will be assuming familiarity with the material in 131AH; some
lecture notes on that material can be found here