Syllabus

The following topics will be surely covered in the days to come. The detailing of these notions will require to have a continuous reading and discussion with me. Remember that a problem in analysis is actually a theorem and not feeding data in a given formula - Happy learning'.
compactness and its characterizations, Total boundedness (The proof of the result pertaining to its characterization may or may not be discussed depending upon how much do we require), completeness of R w.r.t. the usual metric (in other words we will prove that Cauchy convergence if and only iff Convergence when we work with the usual metric), , Definition of limits of several kind (in the language of epsilon-delta), Continuity and differentiability (primarily to make you unlearn the dogmatic ideas of solving multiple choice questions learnt in coaching institutes and take up the deeper concepts of it. Few surprising results waiting for you here), Uniform continuity (no algorithm exists to check whether a function is uniformly continuous or not. We have to understand the definition and negation of a function being continuous in the first place), Sequence of functions - Pointwise convergence, Uniform convergence (including the M-test), (Many results on this topic that will be discussed will involve the knowing of the definitions and the negations thoroughly), Few interesting results will be discussed (without proofs), Definition of norm - Its relation with metric, Completeness of C^0[0,1] w.r.t the ||.||_{sup} norm, Examples of norms w.r.t. which the space C^0[0,1] is incomplete (If time permits we shall see that the space of polynomials is dense subset of C^0[0,1]. Precisely you unlearn the notion of Taylor series expansion here.)

To understand the key aspects of Real Analysis. The keyword is "understand". Mugging up has landed people in this course into deep trouble.

The course will enable a student to 'write' mathematics and not just 'read' it as they have been doing all this while.

S. Kumaresan, Topology of metric Space, Narosa publishing House

Tom Apostol, Calculus: On variable calculus with an introduction to LA, Wiley

Walter Rudin, Principles of Mathematical Analysis, TMH

S.K. Mapa, Introduction to Real Analysis, Levant Books