640:502 Real Analysis II (Spring, 2007) -- Syllabus

Text: Gerald B. Folland, 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
1/17 1 4.1-2 Topological spaces; Continuous maps
1/22 2 4.3-4 Nets; Compact spaces
1/24 3 4.7 Stone-Weierstrass theorem
1/29 4 5.1 Normed vector spaces; Banach spaces
1/31 5 5.2 Bounded linear maps; Dual spaces
2/05 6 5.2 Hahn-Banach theorem
2/07 7 5.3 Baire category theorem; Open mapping theorem
2/12 8 5.3; 5.5 Uniform boundedness principle; Hilbert spaces
2/14 9 5.5 Orthogonal Projections
2/19 10 5.5 Orthonormal bases
2/23 11 6.1 Lp spaces
2/26 12 6.2 Duals of Lp spaces
2/28 13 6.3, 5.4 Integral operators on Lp spaces; Weak convergence
3/05 14 5.4 Topological vector spaces
3/07 15 Midterm Exam (closed book)
3/19 16 8.1 Schwartz space; Lp continuity of translations
3/21 17 8.2 Convolutions; Approximate identities
3/26 18 8.3 Functions on the n-Torus; Fourier series
3/28 19 8.3 Fourier series of smooth functions; Fourier transform
4/02 20 8.3 Fourier inversion formula; Plancherel formula
4/04 21 8.3 Poisson summation formula
4/09 22 8.4 Summability of Fourier series
4/11 23 8.5 Pointwise convergence of Fourier series; Gibb's Phenomenon
4/16 24 7.1 Radon measures and positive linear functionals on C(X)
4/18 25 7.1 Riesz Representation Theorem
4/23 26 7.2 Regularity of Radon measures; Lusin's theorem
4/25 27 7.3 Jordan decomposition; Dual of C(X)
4/30 28 7.3 M(X) and vague convergence of measures
5/08 4-7 pm Final Exam (closed book)

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Roe Goodman / goodman "at" math "dot" rutgers "dot" edu / Revised April 3, 2007