Real Analysis Ii

Riemann integral, integrability, fundamental theorem of calculus, improper integrals. Equicontinuous family of functions, Stone-Weierstrass Theorem. Fourier series. Introduction to metric spaces, the normed space Rn, and in the context of metric spaces the notions of limits, continuity, balls, neighbourhoods, open and closed sets, completeness, compactness, Heine-Borel Theorem, connectedness, space of continuous functions, contraction mapping principle. The Arzela-Ascoli Theorem, Weierstrass Approximation Theorem, the Inverse Function Theorem in Rn, the Implicit Function Theorem in Rn.