Real Analysis II

This is the follow-up course to Real Analysis I. This time we deal with differentiation and integration of functions of several variables. First, we set the stage by studying metric spaces with special emphasis on normed vector spaces. Even here we will encounter several deep theorems like the existence of the completion of metric space, the Arzela—Ascolli theorem as well as the famous Stone—Weierstrass theorem.We will then study the derivative as a linear map and prove the famous implicit and inverse function theorems. These theorems will naturally lead on to the definition of a manifold. We will use the language of manifolds to make precise the method of Lagrange multipliers for constrained optimization.Finally, we will take an elementary approach to the Lebesgue integral that bypasses the more abstract and set-theoretic construction via measures. We will prove all the famous convergence theorems. We will also briefly see how our elementary construction can also be quickly obtained using the completion theorems we studied in metric spaces. The final theorem of the course is the difficult Jacobi transformation formula commonly known as change of variables for which we will give a geometric proof.This course is designed for ambitious undergraduate students as well as beginning graduate students in mathematics. Knowledge of the content of Real Analysis I is assumed as well as content of a basic course in Linear Algebra at the undergraduate level. I will also assume the basics of an undergraduate level course on multivariable calculus as typically done in the first year of BSc./B.Tech.INTENDED AUDIENCE : Anyone Interested LearnersPRE-REQUISITES : Real Analysis I, Linear Algebra, Multivariable CalculusINDUSTRY SUPPORT : DeepMind, Microsoft Research, OpenAI

Week 1:Metric spaces and normed linear spaces Week 2:Compactness and completeness in metric spaces Week 3:The Arzelà–Ascoli and Stone--Weierstrass theorems Week 4:The derivative in several variables as a linear map I Week 5:The derivative in several variables as a linear map II Week 6:The inverse and implicit function theorems Week 7:Manifolds, Tangent spaces and Lagrange multipliers Week 8:A precis of curves and surfaces Week 9:The definition of the Lebesgue integral Week 10:Convergence theorems for the Lebesgue integral Week 11:Multiple Lebesgue Integrals Week 12:The Jacobi transformation formula for Lebesgue integrals