MTH301A - Analysis- I

 Real Number system: Completeness property. Countable and Uncountable. Metric Spaces: Metric spaces, Examples: lp, C[a; b]; Limit, Open sets, Convergence of a sequence, Closed sets, Continuity. Completeness: Complete metric space, Nested set theorem, Baire category theorem, An application. Compactness: Totally bounded, Characterizations of compactness, Finite intersection property, Continuous functions on compact sets, Uniform continuity. Connectedness: Characterizations of connectedness, Continuous functions on connected sets, Path connected. Riemann integration: Denition and existence of integral, Fundamental theorem of calculus. Set of measure zero, Cantor set, Characterization of integrable functions. Convergence of sequence and series of functions: Point wise and uniform convergence of functions, Series of functions, Power series, Dini's theorem, Ascoli's theorem, Continuous function which is no where dierentiable, Weierstrass approximation theorem.