Lecture 59 of 74Week 10
28.1 Levi monotone convergence theorem for step functions
Prof. Jaikrishnan J · IIT Palakkad
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- Real Analysis II, IIT Palakkad (at NPTEL)
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All the lectures
- 1.2 Metric Spaces
- 1.3 Examples of metric spaces
- 1.4 Loads of definitions
- 2.1 Normed vector spaces
- 2.2 Examples of normed vector spaces
- 2.3 Basic properties open closed sets metric
- 3.1 Continuity in metric spaces
- 3.2 Equivalent metrics and product spaces
- 4.1 Completeness
- 4.2 Completeness continued
- 4.3 Completeness of B(x,y)
- 5.1 Completion
- 5.2 Compactness
- 6.1 The Bolzano--Weierstrass Property
- 6.2 Open covers and Compactness
- 6.3 The Heine--Borel Theorem for Metric Spaces
- 7.1 Connectedness
- 7.2 Path-Connectedness
- 7.3 Connected Components
- 8.1 The Arzela--Ascolli theorem
- 8.2 Upper and lower limits
- 9.1 The Stone--Weierstrass theorem
- 9.2 All norms are equivalent
- 10.1 Vector-valued functions
- 10.2 Scalar-valued functions of a vector variable
- 10.3 Directional derivatives and the gradient
- 11.1 Interpretation and properties of the gradient
- 11.2 Higher-order partial derivatives
- 12.1 The derivative as a linear map
- 12.2 Examples of differentiation
- 13.1 Properties of the derivative map
- 13.2 The mean-value theorem
- 13.3 Differentiating under the integral sign
- 14.1 Higher-order derivatives
- 14.2 Symmetry of the second derivative
- 15.1 Taylor's theorem
- 15.2 Taylor's theorem with remainder
- 16.1 The Banach fixed point theorem
- 16.2 Newton's method
- 17.1 The inverse function theorem
- 18.1 Diffeomorphismsm and local diffeomorphisms
- 18.2 The implicit function theorem
- 19.1 Tangent space to a hypersurface
- 20.1 The definition of a manifold
- 21.1 Examples and non examples of manifolds
- 21.2 The tangent space to a manifold
- 22.1 Maxima and minima in several variables
- 22.2 The Hessian and extrema
- 22.3 Completing the squares
- 22.4 Constrained extrema and lagrange multipliers
- 23.1 Curves
- 24.1 Rectifiability and arc-length
- 25.1 The Riemann integral revisited
- 25.2 Monotone sequences of functions
- 26.1 Upper functions and their integrals
- 26.2 Riemann integrable functions as upper functions
- 27.1 Lebesgue integrable functions
- 27.2 Approximation of Lebesgure integrable functions
- 28.1 Levi monotone convergence theorem for step functions
- 28.2 Monotone convergence theorem for upper functions
- 28.3 Monotone convergence theorem for Lebesgue integrable functions
- 29.1 The Lebesgue dominated convergence theorem
- 29.2 Applications of the convergence theorems
- 30.1 The problem of measure
- 31.1. The Lebesgue integral on unbounded intervals
- 31.2 Measurable functions
- 32.1 Solution to the problem of measure
- 33.1 The Lebesgue integral on arbitrary subsets
- 33.2 Square integrable functions
- 34.1 Norms and inner-products on complex vector spaces
- 34.2 Convergence in L2
- 34.3 The Riesz--Fischer theorem
- 35.1 Multiple Riemann integration
- 35.2 Multiple Lebesgue integration