MAT 288E - Real Analysis I

Course Objectives

1. To teach the student the techniques and methods of Mathematical Analysis and to allow the student to develop a certain level of proficiency in these methods.
2. To teach students to use the basic concepts they learned in Calculus classes in a mathematically rigourous way.

Course Description

Sequences of real numbers and convergence. Convergence.. Normed vector spaces. Finite dimensional real vector spaces. Young’s, Hölder’s and Minkowski’s inequalities. Metric spaces. Sequences in metric spaces. Convergence and boundedness. Cauchy sequences and completeness. Topology of Metric spaces: open and closed sets. Compactness. Heine-Borel Theorem. Real valued continuous functions on metric spaces and their metric structure. Hilbert spaces. Continuity and uniform continuity. Lipschitz continuity. Total Derivative. Lebesgue Measure on R. Lebesgue measurable functions. Lebesgue Integration on R. Normed spaces C^k[a,b], \ell^p and L^p. Sequences and series of real valued functions on metric spaces. The Stone-Weierstrass Theorem. Pointwise and uniform convergence. Cauchy criterion for uniform convergence. Weierstrass M-test.