Text: Gerald B. Foland, Real Analysis: Modern
Techniques and Their Applications (2nd ed.),
ISBN #0-471-31716-0, Wiley-Interscience/John Wiley Sons, Inc., 1999.
| Date | Lecture | Reading | Topics |
|---|---|---|---|
| 1/21 | 1 | 4.1-2 | Topological spaces; Continuous maps |
| 1/26 | 2 | 4.3-4 | Nets; Compact spaces |
| 1/28 | 3 | 4.7 | Stone-Weierstrass theorem |
| 2/02 | 4 | 5.1 | Normed vector spaces; Banach spaces |
| 2/04 | 5 | 5.2 | Bounded linear maps; Dual spaces |
| 2/09 | 6 | 5.3 | Baire category theorem; Open mapping theorem |
| 2/11 | 7 | 5.3 | Uniform boundedness principle |
| 2/16 | 8 | 5.5 | Hilbert spaces; Orthogonal Projections |
| 2/18 | 9 | 5.5 | Hilbert spaces; Orthonormal bases |
| 2/23 | 10 | 6.1 | Lp spaces |
| 2/25 | 11 | 6.2 | Duals of Lp spaces |
| 3/01 | 12 | 6.3 | Integral operators on Lp spaces; Convolution operators |
| 3/03 | 13 | 5.4 | Weak convergence; Topological vector spaces |
| 3/08 | 14 | 5.4 | Frechet spaces |
| 3/10 | 15 | Midterm Exam | (closed book) |
| 3/22 | 16 | 8.1 | Schwartz space; Lp continuity of translations |
| 3/24 | 17 | 8.2 | Convolutions; Approximate identities |
| 3/29 | 18 | 8.3 | Functions on the n-Torus; Fourier series |
| 3/31 | 19 | 8.3 | Fourier series of smooth functions; Fourier transform |
| 4/05 | 20 | 8.3 | Fourier inversion formula; Plancherel formula |
| 4/07 | 21 | 8.3 | Poisson summation formula |
| 4/12 | 22 | 8.4 | Summability of Fourier series |
| 4/14 | 23 | 8.5 | Pointwise convergence of Fourier series; Gibb's Phenomenon |
| 4/19 | 24 | 7.1 | Radon measures and positive linear functionals on C(X) |
| 4/21 | 25 | 7.1 | Riesz Representation Theorem |
| 4/26 | 26 | 7.2 | Regularity of Radon measures; Lusin's theorem |
| 4/28 | 27 | 7.3 | Jordan decomposition; Dual of C(X) |
| 5/03 | 28 | 7.3 | M(X) and vague convergence of measures |
| 5/10 | 4-7 pm | Final Exam | (closed book) |
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