NPTEL : NOC:Real Analysis - II (Mathematics)

Co-ordinators : Prof. Jaikrishnan J


Lecture 1 - Metric Spaces

Lecture 2 - Examples of metric spaces

Lecture 3 - Loads of definitions

Lecture 4 - Normed vector spaces

Lecture 5 - Examples of normed vector spaces

Lecture 6 - Basic properties open closed sets metric

Lecture 7 - Continuity in metric spaces

Lecture 8 - Equivalent metrics and product spaces

Lecture 9 - Completeness

Lecture 10 - Completeness (Continued...)

Lecture 11 - Completeness of B(x,y)

Lecture 12 - Completion

Lecture 13 - Compactness

Lecture 14 - The Bolzano-Weierstrass Property

Lecture 15 - Open covers and Compactness

Lecture 16 - The Heine-Borel Theorem for Metric Spaces

Lecture 17 - Connectedness

Lecture 18 - Path-Connectedness

Lecture 19 - Connected Components

Lecture 20 - The Arzela-Ascolli theorem

Lecture 21 - Upper and lower limits

Lecture 22 - The Stone-Weierstrass theorem

Lecture 23 - All norms are equivalent

Lecture 24 - Vector-valued functions

Lecture 25 - Scalar-valued functions of a vector variable

Lecture 26 - Directional derivatives and the gradient

Lecture 27 - Interpretation and properties of the gradient

Lecture 28 - Higher-order partial derivatives

Lecture 29 - The derivative as a linear map

Lecture 30 - Examples of differentiation

Lecture 31 - Properties of the derivative map

Lecture 32 - The mean-value theorem

Lecture 33 - Differentiating under the integral sign

Lecture 34 - Higher-order derivatives

Lecture 35 - Symmetry of the second derivative

Lecture 36 - Taylor's theorem

Lecture 37 - Taylor's theorem with remainder

Lecture 38 - The Banach fixed point theorem

Lecture 39 - Newton's method

Lecture 40 - The inverse function theorem

Lecture 41 - Diffeomorphismsm and local diffeomorphisms

Lecture 42 - The implicit function theorem

Lecture 43 - Tangent space to a hypersurface

Lecture 44 - The definition of a manifold

Lecture 45 - Examples and non examples of manifolds

Lecture 46 - The tangent space to a manifold

Lecture 47 - Maxima and minima in several variables

Lecture 48 - The Hessian and extrema

Lecture 49 - Completing the squares

Lecture 50 - Constrained extrema and lagrange multipliers

Lecture 51 - Curves

Lecture 52 - Rectifiability and arc-length

Lecture 53 - The Riemann integral revisited

Lecture 54 - Monotone sequences of functions

Lecture 55 - Upper functions and their integrals

Lecture 56 - Riemann integrable functions as upper functions

Lecture 57 - Lebesgue integrable functions

Lecture 58 - Approximation of Lebesgure integrable functions

Lecture 59 - Levi monotone convergence theorem for step functions

Lecture 60 - Monotone convergence theorem for upper functions

Lecture 61 - Monotone convergence theorem for Lebesgue integrable functions

Lecture 62 - The Lebesgue dominated convergence theorem

Lecture 63 - Applications of the convergence theorems

Lecture 64 - The problem of measure

Lecture 65 - The Lebesgue integral on unbounded intervals

Lecture 66 - Measurable functions

Lecture 67 - Solution to the problem of measure

Lecture 68 - The Lebesgue integral on arbitrary subsets

Lecture 69 - Square integrable functions

Lecture 70 - Norms and inner-products on complex vector spaces

Lecture 71 - Convergence in L2

Lecture 72 - The Riesz-Fischer theorem

Lecture 73 - Multiple Riemann integration

Lecture 74 - Multiple Lebesgue integration