Course Description: | This is a course in real analysis for students who have already met the basic concepts of sequences and continuity on the real line. This course generalizes these concepts to Euclidean spaces and to more general metric and normed spaces. These more general spaces are introduced at the start and are emphasized throughout the course. Topics include sequences and series on the real line; metric and normed spaces; open and closed sets, topological properties of sets and equivalent metrics, sequences in metric spaces, compactness, and completeness; continuity of real valued functions and of functions between metric spaces, uniform continuity, and Lipschitz condition; differentiation of real valued functions, the mean value theorem, differentiation of functions between Euclidean spaces, and partial derivatives; Riemann integral and the fundamental theorem of calculus; sequences and series of functions; pointwise and uniform convergence of sequences of functions, power series, and series in normed spaces. |