Courses

Sequences of functions, uniform convergence and its metric, spaces of continuous functions. Dini's theorem. Alternating limit processes (derivation, integration) in the presence of uniform convergence. Series of functions, criteria for uniform convergence (M-test, Abel, Dirichlet). Fourier analysis, Bessel inequality, Parseval identity. Compact subsets in metric spaces, images through continuous functions. Characterization of compact subsets of ℝn. Stone - Weierstrass theorem. Connected subsets in metric spaces, images through continuous functions. Characterization of connected subsets of ℝ.