Kuttler K. - Basic Analysis

Kuttler K. - Basic Analysis - Basic Analysis Kenneth...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Basic Analysis Kenneth Kuttler April 16, 2001 2 Contents 1 Basic Set theory 9 1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 The Schroder Bernstein theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Linear Algebra 15 2.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 The characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 The rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 General topology 43 3.1 Compactness in metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 The Tychonoff theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Spaces of Continuous Functions 61 4.1 Compactness in spaces of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Stone Weierstrass theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Abstract measure and Integration 71 5.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Monotone classes and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4 The Abstract Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 The space L 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.6 Double sums of nonnegative terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.7 Vitali convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.8 The ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
View Full Document

This note was uploaded on 08/04/2010 for the course DMA Diversos taught by Professor Diversos during the Spring '10 term at University of Florida.

Page1 / 499

Kuttler K. - Basic Analysis - Basic Analysis Kenneth...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online