Schedule

Week 1 (May 20): Preliminary Review
- Uniform convergence, M-Test
- 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, Cauchy-Schwarz)
- 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
- Borel-Cantelli
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)
- Fubini-Tonelli 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:
- \(L^1\), \(L^2\), \(L^\infty\), Cauchy-Schwarz, Pythagoras for orthogonality
- Function spaces
- Plancherel, Parseval, Bessel, Riesz Representation
Week 8 (July 8): Abstract Measures
- Radon-Nikodym

References