Schedule

Week 1 (May 20): Preliminary Review
 Uniform convergence, MTest
 Nowhere density, Baire category, Heine Borel
 Normed spaces
 Series and sequences, convergence, small tails, limsup and liminf, Cauchy criteria for sums and integrals
 Basic inequalities (triangle, CauchySchwarz)
 Tools from Calculus: MVT, Taylor’s theorem & remainder
 Weierstrass approximation
 Pathological functions and continuity

Week 2 (May 27): Measure Theory (Sets)
 \(F_\sigma\) and \(G_\delta\) sets
 Sigma algebras
 Outer measure, Lebesgue measure
 Continuity of measure
 BorelCantelli

Week 3 (June 3): Integration 1:
 Measurable Functions
 Lebesgue integration, change of variables
 Chebyshev’s inequality
 Egorov, Lusin
 Types of convergence: uniform, pointwise, a.e., in measure, in norm

Week 4 (June 10): Integration 2:
 Convergence Theorems (Fatou, Monotone, Dominated)
 FubiniTonelli and repeated integration

Week 5 (June 17): Fourier Analysis
 The Fourier transform
 Fourier series
 Convolutions, approximations to the identity
 Trigonometric series, density of trig polynomials

Week 6 (June 24): Functional Analysis 1
 Banach and Hilbert Spaces, completeness
 l^p and L^p
 Holder
 Duals, linear functionals, operator norm

Week 7 (July 1): Functional Analysis 2:
 \(L^1\), \(L^2\), \(L^\infty\), CauchySchwarz, Pythagoras for orthogonality
 Function spaces
 Plancherel, Parseval, Bessel, Riesz Representation

Week 8 (July 8): Abstract Measures
 RadonNikodym
References

Topics
 Convergence theorems for integrals, Borel measure, Riesz representation theorem
 L space, Duality of L space, Jensen inequality
 Lebesgue differentiation theorem, Fubini theorem, Hilbert space
 Complex measures of bounded variation, RadonNikodym theorem.
 Fourier series, Fourier transform, convolution.
 Heat equation, Dirichlet problem, fundamental solutions
 Central limit theorem, law of large numbers, conditional probability and conditional expectation.
 Distributions, Sobolev embedding theorem.
 Maximum principle.