MAT 272E - Advanced Mathematics

Course Objectives

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.
To teach students to use the basic concepts they learned in Calculus classes in a mathematically rigourous way.

Course Description

Sequences and Series of Real Numbers and Convergence. 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. Continuity and Uniform Continuity. Lipschitz Continuity. Total Derivative. C^k[a,b] and ell^p spaces. Sequences and Series of Real Valued Functions on Metric Spaces. Pointwise and Uniform Convergence. Cauchy Criterion for Uniform Convergence. Weierstrass M-test. The Stone-Weierstrass Theorem. Hilbert Spaces.