Real Analysis I, II - 60350, 60360

1. Calculus
Calculus of one and several variables, the Implicit and Inverse Function Theorems, pointwise and uniform convergence of sequences of functions, integration and differentiation of sequences, the Weierstrass Approximation Theorem.

2. Lebesgue measure and integration on the real line
Measurable sets, Lebesgue measure, measurable functions, the Lebesgue integral and its relation to the Riemann integral, convergence theorems, functions of bounded variation, absolute continuity and differentiation of integrals.

3. General measure and integration theory
Measure spaces, measurable functions, integration convergence theorems, signed measures, the Radon-Nikodym Theorem, product measures, Fubini’s Theorem, Tonelli’s Theorem.

4. Families of functions
Equicontinuous families and the Arzela-Ascoli Theorem, the Stone-Weierstrass Theorem.

5. Banach spaces
L
P- spaces and their conjugates, the Riesz-Fisher Theorem, the Riesz Representation Theorem for bounded linear functionals on LP, C(X), the Riesz Representation Theorem for C(X), the Hahn-Banach Theorem, the Closed Graph and Open Mapping Theorems, the Principle of Uniform Boundedness, Alaoglu’s Theorem, Hilbert spaces, orthogonal systems, Fourier series, Bessel’s inequality, Parseval’s formula, convolutions, Fourier transform, distributions, Sobolev spaces.

References
Apostol, Mathematical Analysis.
Knapp, Basic Real Analysis. Riesz-Nagy, Functional Analysis. Royden, Real Analysis.
Rudin, Principles of Mathematical Analysis.
Rudin, Real and Complex Analysis.
Rudin, Functional Analysis.
Simmons, Introduction to Topology and Modern Analysis.
Wheeden-Zygmund, Measure and Integration.
Folland, Real Analysis.

Real Analysis I covers the material on calculus, and Lebesgue measure and integration. It is roughly Chapters I-III and V- VI of Knapp. The remaining material is in Real Analysis II.