I will be taking analysis course in a month from now. Topics are given below. I am doing engineering and had been through calculus courses but nothing like sort of analysis before. Many of my friends who are engineers themselves have warned me about the difficulty and abstraction level of this course. Can some one suggest what should i do in order to learn it properly. I will be sitting with maths majors and donot want to look dumb. Any books for self study or whatever will be appreciated.Any tips or help thanks..
MAL411 Topics in REAL ANALYSIS, 4 (3-1-0)
Course contents : Metric spaces, completeness, connectedness, compactness, Heine-Borel theorem, totally bounded sets, finite intersection property, completeness of R^n, Banach fixed point theorem, perfect sets, the Cantor set. Continuous functions, relation with connectedness and compactness, discontinuity, uniform continuous functions, monotone functions, Absolutely continuous functions, total variation and functions of bounded variations. Differentiability and its properties, mean value theorem, Taylor's theorem, Riemann integral with properties and characterization, improper integral, Gamma function, Directional derivative, Partial derivative, Derivative as a linear transformation, Inverse and Implicit function theorems, multiple integration, Change of variables. Sequence and series of real numbers, point wise convergence, Fejer's theorem, power series and Fourier series, uniform convergence and its relation with continuity, differentiability and inerrability, Weierstrass approximation theorem, Equi-continuous family, Arzela-Ascoli theorem.