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Lecture notes Kenneth L. Kuttler Multivariable Advanced Calculus

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Lecture notes Kenneth L. Kuttler Multivariable Advanced Calculus Lecture notes Kenneth L. Kuttler Multivariable Advanced Calculus Lecture notes Kenneth L. Kuttler Multivariable Advanced Calculus Lecture notes Kenneth L. Kuttler Multivariable Advanced Calculus Lecture notes Kenneth L. Kuttle...

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  • December 29, 2022
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Lecture notes
Kenneth L. Kuttler
Multivariable
Advanced
Calculus 2020.

,Multivariable Advanced Calculus

Kenneth Kuttler

February 7, 2016

,2

, Contents

1 Introduction 9

2 Some Fundamental Concepts 11
2.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 The Schroder Bernstein Theorem . . . . . . . . . . . . . . . . . . 13
2.1.3 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 lim sup And lim inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Double Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Basic Linear Algebra 25
3.1 Algebra in Fn , Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Subspaces Spans And Bases . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Block Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1 The Determinant Of A Matrix . . . . . . . . . . . . . . . . . . . 39
3.5.2 The Determinant Of A Linear Transformation . . . . . . . . . . 50
3.6 Eigenvalues And Eigenvectors Of Linear Transformations . . . . . . . . 51
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.8 Inner Product And Normed Linear Spaces . . . . . . . . . . . . . . . . . 54
3.8.1 The Inner Product In Fn . . . . . . . . . . . . . . . . . . . . . . 54
3.8.2 General Inner Product Spaces . . . . . . . . . . . . . . . . . . . . 55
3.8.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8.4 The p Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8.5 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.8.6 The Adjoint Of A Linear Transformation . . . . . . . . . . . . . 61
3.8.7 Schur’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.9 Polar Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Sequences 73
4.1 Vector Valued Sequences And Their Limits . . . . . . . . . . . . . . . . 73
4.2 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Closed And Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Cauchy Sequences And Completeness . . . . . . . . . . . . . . . . . . . 81
4.5 Shrinking Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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