Real Analysis II RDXR89VKTQ

URI	`m2://res/RDXR89VKTQ`
Kind	course
Creators	Jaikrishnan J (instructor)
Date	2021-06-08
Scope	in

Expressions

e1: edition

Language	en
Date	2021-06-08

m1: page, IIT Palakkad (record). nptel 111106153
m2: page, 2021-06-08 (record). nptel 111106153

Contents

part 1: Week 1
session 1: 1.2 Metric Spaces
session 2: 1.3 Examples of metric spaces
session 3: 1.4 Loads of definitions
session 4: 2.1 Normed vector spaces
session 5: 2.2 Examples of normed vector spaces
session 6: 2.3 Basic properties open closed sets metric
session 7: 3.1 Continuity in metric spaces
session 8: 3.2 Equivalent metrics and product spaces
part 2: Week 2
session 9: 4.1 Completeness
session 10: 4.2 Completeness continued
session 11: 4.3 Completeness of B(x,y)
session 12: 5.1 Completion
session 13: 5.2 Compactness
session 14: 6.1 The Bolzano--Weierstrass Property
session 15: 6.2 Open covers and Compactness
session 16: 6.3 The Heine--Borel Theorem for Metric Spaces
part 3: Week 3
session 17: 7.1 Connectedness
session 18: 7.2 Path-Connectedness
session 19: 7.3 Connected Components
session 20: 8.1 The Arzela--Ascolli theorem
session 21: 8.2 Upper and lower limits
session 22: 9.1 The Stone--Weierstrass theorem
session 23: 9.2 All norms are equivalent
part 4: Week 4
session 24: 10.1 Vector-valued functions
session 25: 10.2 Scalar-valued functions of a vector variable
session 26: 10.3 Directional derivatives and the gradient
session 27: 11.1 Interpretation and properties of the gradient
session 28: 11.2 Higher-order partial derivatives
session 29: 12.1 The derivative as a linear map
session 30: 12.2 Examples of differentiation
part 5: Week 5
session 31: 13.1 Properties of the derivative map
session 32: 13.2 The mean-value theorem
session 33: 13.3 Differentiating under the integral sign
session 34: 14.1 Higher-order derivatives
session 35: 14.2 Symmetry of the second derivative
session 36: 15.1 Taylor's theorem
session 37: 15.2 Taylor's theorem with remainder
part 6: Week 6
session 38: 16.1 The Banach fixed point theorem
session 39: 16.2 Newton's method
session 40: 17.1 The inverse function theorem
session 41: 18.1 Diffeomorphismsm and local diffeomorphisms
session 42: 18.2 The implicit function theorem
part 7: Week 7
session 43: 19.1 Tangent space to a hypersurface
session 44: 20.1 The definition of a manifold
session 45: 21.1 Examples and non examples of manifolds
session 46: 21.2 The tangent space to a manifold
part 8: Week 8
session 47: 22.1 Maxima and minima in several variables
session 48: 22.2 The Hessian and extrema
session 49: 22.3 Completing the squares
session 50: 22.4 Constrained extrema and lagrange multipliers
session 51: 23.1 Curves
session 52: 24.1 Rectifiability and arc-length
part 9: Week 9
session 53: 25.1 The Riemann integral revisited
session 54: 25.2 Monotone sequences of functions
session 55: 26.1 Upper functions and their integrals
session 56: 26.2 Riemann integrable functions as upper functions
session 57: 27.1 Lebesgue integrable functions
session 58: 27.2 Approximation of Lebesgure integrable functions
part 10: Week 10
session 59: 28.1 Levi monotone convergence theorem for step functions
session 60: 28.2 Monotone convergence theorem for upper functions
session 61: 28.3 Monotone convergence theorem for Lebesgue integrable functions
session 62: 29.1 The Lebesgue dominated convergence theorem
session 63: 29.2 Applications of the convergence theorems
session 64: 30.1 The problem of measure
part 11: Week 11
session 65: 31.1. The Lebesgue integral on unbounded intervals
session 66: 31.2 Measurable functions
session 67: 32.1 Solution to the problem of measure
session 68: 33.1 The Lebesgue integral on arbitrary subsets
session 69: 33.2 Square integrable functions
part 12: Week 12
session 70: 34.1 Norms and inner-products on complex vector spaces
session 71: 34.2 Convergence in L2
session 72: 34.3 The Riesz--Fischer theorem
session 73: 35.1 Multiple Riemann integration
session 74: 35.2 Multiple Lebesgue integration

Catalogue records

nptel: 111106153 (manifestation)
nptel_video: 111106153 (manifestation)