Real Analysis

So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn’t that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist’s eye? Far from that, without this language most of the intimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality.

 Henri Poincaré

Real Analysis is an area of mathematics that was developed to formalize the study of numbers and functions and to investigate important concepts such as limits and continuity. These concepts underpin calculus and its applications. Real Analysis has become an indispensable tool in a number of application areas. In particular, many of its key concepts, such as convergence, compactness and convexity, have become central to economic theory.

To understand Real Analysis to a professional extent you need the following prerequisites-

  1. Multivariate Calculus – Intermediate Level
  2. Linear Algebra – Intermediate Level
  3. Methods of Proofs
  4. Basic Set Theory

Key Points to be focused in real Analysis-

  • Basics: proof, logic, sets and functions
  • Real numbers and sequences
  • Functions, limits and continuity
  • Infinite series
  • Metric and normed spaces
  • Convergence, completeness and compactness
  • Continuity in metric spaces
  • The derivative
  • Convexity
  • Fixed point theorems

This series shall be divided into short chapters for better understanding and retention.