Analysis

Chapter 6
Implicit Function Theorem

Part August 25, 2018 I  Topology, Continuity, And Algebra

Chapter 1 Some Prerequisite Material

1.1 Basic Definitions
1.2 The Schroder Bernstein Theorem
1.3 Equivalence Relations
1.4 sup And inf
1.5 Double Series
1.6 lim sup and lim inf

Chapter 2 Metric Spaces

2.1 Open And Closed Sets, Sequences, Limit Points, Completeness
2.2 Cauchy Sequences
2.3 Closure Of A Set
2.4 Separable Metric Spaces
2.5 Compact Sets
2.6 Continuous Functions
2.7 Continuity And Compactness
2.8 Lipschitz Continuity And Contraction Maps
2.9 Convergence Of Functions
2.10 Compactness In C (X,Y ) Ascoli Arzela Theorem∗
2.11 Connected Sets
2.12 Exercises

Chapter 3 Normed Linear Spaces

3.1 Algebra in Fn, Vector Spaces
3.2 Subspaces Spans And Bases
3.3 Inner Product And Normed Linear Spaces
3.3.1 The Inner Product In Fn
3.3.2 General Inner Product Spaces
3.3.3 Normed Vector Spaces
3.3.4 The p Norms
3.3.5 Orthonormal Bases
3.4 Equivalence Of Norms
3.5 Exercises
3.6 ℒ(V,W) As A Vector Space
3.7 The Norm Of A Linear Map, Operator Norm
3.8 Continuous Functions In Normed Linear Space
3.9 Polynomials
3.10 Weierstrass Approximation Theorem
3.11 Functions Of Many Variables
3.12 Tietze Extension Theorem
3.13 Connectedness In Normed Linear Space
3.14 Exercises

Chapter 4 Brouwer Fixed Point Theorem ℝn∗

4.1 Simplices And Triangulations
4.2 Labeling Vertices
4.3 The Brouwer Fixed Point Theorem

Part II  Differentiation

Chapter 5 The Derivative

5.1 Limits Of A Function
5.2 Basic Definitions
5.3 The Chain Rule
5.4 The Matrix Of The Derivative
5.5 A Mean Value Inequality
5.6 Existence Of The Derivative, C1 Functions
5.7 Higher Order Derivatives
5.8 Some Standard Notation
5.9 The Derivative And The Cartesian Product
5.10 Mixed Partial Derivatives
5.11 Newton’s Method
5.12 Exercises

Chapter 6 Implicit Function Theorem

6.1 Statement And Proof Of The Theorem
6.2 More Derivatives
6.3 The Case Of ℝn
6.4 Exercises
6.5 The Method Of Lagrange Multipliers
6.6 The Taylor Formula
6.7 Second Derivative Test
6.8 The Rank Theorem
6.9 The Local Structure Of C1 Mappings
6.10 Invariance Of Domain∗
6.11 Exercises

Part III  Integration

Chapter 7 Abstract Measures And Measurable Functions

7.1 Simple Functions And Measurable Functions
7.2 Measures And Their Properties
7.3 Kuratowski’s Lemma
7.4 Dynkin’s Lemma
7.5 Measures And Regularity
7.6 Examples Of Measures And Outer Measures
7.7 Exercises
7.8 An Outer Measure On P (ℝ)
7.9 Measures From Outer Measures
7.10 One Dimensional Lebesgue Stieltjes Measure
7.11 Exercises

Chapter 8 The Abstract Lebesgue Integral

8.1 Definition For Nonnegative Measurable Functions
8.1.1 Riemann Integrals For Decreasing Functions
8.1.2 The Lebesgue Integral For Nonnegative Functions
8.2 The Lebesgue Integral For Nonnegative Simple Functions
8.3 The Monotone Convergence Theorem
8.4 Other Definitions
8.5 Fatou’s Lemma
8.6 The Integral’s Righteous Algebraic Desires
8.7 The Lebesgue Integral, L1
8.8 The Dominated Convergence Theorem
8.9 Exercises
8.10 Some Important Measure Theory
8.10.1 Eggoroff’s Theorem
8.10.2 The Vitali Convergence Theorem
8.11 The One Dimensional Lebesgue Stieltjes Integral
8.12 The Distribution Function
8.13 Good Lambda Inequality

Chapter 9 The Lebesgue Integral In ℝn

9.1 n Dimensional Lebesgue Measure And Integrals
9.1.1 Iterated Integrals
9.1.2 n Dimensional Lebesgue Measure And Integrals
9.1.3 The Sigma Algebra Of Lebesgue Measurable Sets
9.1.4 Fubini’s Theorem
9.2 Approximation With Cc (Y )
9.3 Covering Theorems
9.4 Exercises
9.5 Change Of Variables, Linear Maps
9.6 Differentiable Functions And Measurability
9.7 Change Of Variables, Nonlinear Maps
9.8 The Mapping Is Only One To One
9.9 Mappings Which Are Not One To One
9.10 Spherical Coordinates In p Dimensions
9.11 Brouwer Fixed Point Theorem
9.12 Invariance Of Domain∗
9.13 Exercises

Chapter 10 Integration On Manifolds

10.1 Mollifiers And Partitions Of Unity
10.2 Manifolds
10.3 The Binet Cauchy Formula
10.4 The Area Measure On A Manifold
10.5 Exercises

Chapter 11 Divergence Theorem

11.1 Divergence Theorem
11.2 A Generalization

Chapter 12 Line Integrals

12.0.1 Arc Length And Orientations
12.0.2 Another Notation For Line Integrals
12.1 The Cross Product And Notation
12.1.1 The Box Product And Distributive Law
12.1.2 Proof Of The Distributive Law

Chapter 13 Green’s And Stoke’s Theorems

13.1 Green’s Theorem
13.2 Stoke’s Theorem
13.2.1 The Normal And The Orientation
13.2.2 The Mobeus Band
13.2.3 Conservative Vector Fields
13.2.4 Some Terminology
13.3 Exercises

Chapter 14 Differential Forms

14.1 Basic Considerations
14.2 The Wedge Product
14.3 The Exterior Derivative
14.4 Stoke’s Theorem
14.5 Examples

Chapter 15 The Lp Spaces

15.1 Basic Inequalities And Properties
15.2 Density Considerations
15.3 Separability
15.4 Continuity Of Translation
15.5 Mollifiers And Density Of Smooth Functions
15.6 Exercises

Chapter 16 Degree Theory, An Introduction

16.1 Sard’s Lemma and Approximation
16.2 An Identity and Surprising Implications
16.3 Definitions And Elementary Properties
16.4 Borsuk’s Theorem
16.5 Applications
16.6 The Product Formula and Jordan Separation Theorem
16.7 Jordan Curve Theorem
16.8 Exercises

Part IV  The Integral And The Derivative In ℝn

Chapter 17 Integrals And Derivatives

17.1 The Fundamental Theorem Of Calculus
17.2 Lipschitz Functions
17.3 Rademacher’s Theorem
17.4 Weak Derivatives

Chapter 18 Differentiation Of Radon Measures

18.1 Besicovitch Covering Theorem
18.2 Fundamental Theorem Of Calculus For Radon Measures
18.3 Vitali Coverings
18.4 Differentiation Of Radon Measures
18.5 The Radon Nikodym Theorem For Radon Measures
18.6 Absolutely Continuous Functions

Chapter 19 Hausdorff Measure

19.1 Definition Of Hausdorff Measures
19.1.1 Properties Of Hausdorff Measure
19.2 ℋn And mn
19.3 Technical Considerations
19.3.1 Steiner Symmetrization
19.3.2 The Isodiametric Inequality
19.4 The Proper Value Of β (n)
19.4.1 A Formula For α (n)
19.4.2 Hausdorff Measure And Linear Transformations

Chapter 20 The Area Formula

20.0.1 Preliminary Results
20.0.2 The Area Formula
20.1 Mappings That Are Not One To One
20.2 The Divergence Theorem
20.3 The Coarea Formula
20.4 Change Of Variables

Part V  Abstract Theory

Chapter 21 Hausdorff Spaces And Measures

21.1 General Topological Spaces
21.2 The Alexander Sub-basis Theorem
21.3 Stone Weierstrass Theorem
21.3.1 The Case Of Compact Sets
21.3.2 The Case Of Locally Compact Sets
21.3.3 The Case Of Complex Valued Functions
21.4 Partitions Of Unity
21.5 Measures On Hausdorff Spaces
21.6 Measures And Positive Linear Functionals
21.7 Slicing Measures∗
21.8 Exercises

Chapter 22 Product Measures

22.1 General Theory
22.2 Completion Of Product Measure Spaces
22.3 Product Measures For Infinite Products
22.3.1 Algebras
22.3.2 Caratheodory Extension Theorem
22.4 Kolmogorov Extension Theorem∗
22.5 Exercises

Chapter 23 Banach Spaces

23.1 Theorems Based On Baire Category
23.1.1 Baire Category Theorem
23.1.2 Uniform Boundedness Theorem
23.1.3 Open Mapping Theorem
23.1.4 Closed Graph Theorem
23.2 Hahn Banach Theorem
23.2.1 Partially Ordered Sets
23.2.2 Gauge Functions And Hahn Banach Theorem
23.2.3 The Complex Version Of The Hahn Banach Theorem
23.2.4 The Dual Space And Adjoint Operators
23.3 Uniform Convexity Of Lp
23.4 Closed Subspaces
23.5 Weak And Weak ∗ Topologies
23.5.1 Basic Definitions
23.5.2 Banach Alaoglu Theorem
23.5.3 Eberlein Smulian Theorem
23.6 Exercises

Chapter 24 Hilbert Spaces

24.1 Basic Theory
24.2 The Hilbert Space L(U)
24.3 Approximations In Hilbert Space
24.4 Orthonormal Sets
24.5 Compact Operators
24.5.1 Compact Operators In Hilbert Space
24.5.2 Nuclear Operators
24.5.3 Hilbert Schmidt Operators
24.6 Square Roots
24.7 Ordinary Differential Equations in Banach Space
24.8 Fractional Powers of Operators
24.9 General Theory Of Continuous Semigroups
24.9.1 An Evolution Equation
24.9.2 Adjoints, Hilbert Space
24.9.3 Adjoints, Reflexive Banach Space
24.10 Exercises

Chapter 25 Representation Theorems

25.1 Radon Nikodym Theorem
25.2 Vector Measures
25.3 Representation Theorems For The Dual Space Of Lp
25.4 The Dual Space Of L∞(Ω)
25.5 Non σ Finite Case
25.6 The Dual Space Of C0 (X)
25.7 The Dual Space Of C0(X), Another Approach
25.8 More Attractive Formulations
25.9 Exercises

Chapter 26 Fourier Transforms

26.1 Fourier Transforms Of Functions In G
26.2 Fourier Transforms Of Just About Anything
26.2.1 Fourier Transforms Of G∗
26.2.2 Fourier Transforms Of Functions In L1 (ℝn)
26.2.3 Fourier Transforms Of Functions In L2 (ℝn)
26.2.4 The Schwartz Class
26.2.5 Convolution
26.3 Exercises
A.1 The Matrix Of A Linear Map
A.2 Block Multiplication Of Matrices
A.3 Schur’s Theorem
A.4 Hermitian And Symmetric Matrices
A.5 The Right Polar Factorization
A.6 Elementary matrices
A.7 The Row Reduced Echelon Form Of A Matrix
A.8 Finding The Inverse Of A Matrix
A.9 The Mathematical Theory Of Determinants
A.9.1 The Function sgn
A.9.2 The Definition Of The Determinant
A.9.3 A Symmetric Definition
A.9.4 Basic Properties Of The Determinant
A.9.5 Expansion Using Cofactors
A.9.6 A Formula For The Inverse
A.9.7 Rank Of A Matrix
A.9.8 Summary Of Determinants
A.10 The Cayley Hamilton Theorem
B.1 The Hamel Basis
C.1 Partitions Of Unity And Stone’s Theorem
C.2 An Extension Theorem, Retracts
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