608 Notes - Real Analysis MATH - 607/8 Contents Contents 1...

Info iconThis preview shows pages 1–5. 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: Real Analysis MATH - 607/8 Contents Contents 1 Contents 1 1 General Topology 3 1.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Continuous Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Two Compactness Theorems . . . . . . . . . . . . . . . . . . . 18 1.6 The Stone-Weierstrass Theorem . . . . . . . . . . . . . . . . . . 21 2 Elements of Functional Analysis 25 2.1 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Construction of Normed Linear Spaces . . . . . . . . . . . . . . 27 2.3 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 The Dual of L p . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5 The Baire Category Theorem . . . . . . . . . . . . . . . . . . . 44 2.6 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . 48 2.7 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1 2 Contents 3 Radon Measures 59 3.1 Positive Linear Functionals on C c ( X ) . . . . . . . . . . . . . . . 60 Index 67 Chapter 1 General Topology Let ( X,d ) be a metric space i.e., d : X X [0 , ) such that 1. d ( x,x ) = 0 , x X 2. d ( x,y ) = 0 x = y, x,y X 3. d ( x,y ) + d ( y,z ) d ( x,z ) , x,y,z X 4. d ( x,y ) = d ( y,x ) , x,y X Remark. U X is open if x U > B ( x ) U . F X is closed if F c is open. Open sets are closed under taking arbitrary unions and finite intersection. We would like to extend the notion of open and closed sets to general spaces. 1.1 TOPOLOGICAL SPACES Definition 1.1 (Topology) . Let X be a set. T P ( X ) is a topology of X if 1. ,X T , 2. ( U j ) j J T uniondisplay j J U j T , for an arbitrary index set J and 3. ( U j ) j J T intersectiondisplay j J U j T , for any finite index set J . The elements of the topology are called open sets and ( X, T ) is called a Topological Space. When there is no ambiguity, we refer to X as a topological space. F X is closed if X \ F is open. Example 1.2 (Topologies and Topological Spaces) . 1. Discrete Topology - T = P ( X ). 2. Indiscrete (Trivial) Topology - T = { ,X } . 3 4 CHAPTER 1. GENERAL TOPOLOGY 3. Cofinite Topology - T = { A X | X \ A is finite }{ } . Here A is closed A = X or A is finite. 4. All metric spaces are topological spaces. Definition 1.3. ( X, T ) is a topological space. Let A X , then A = uniondisplay U A U open U is called the open kernel of A , and A = intersectiondisplay F A F closed F is the closure of A ....
View Full Document

Page1 / 68

608 Notes - Real Analysis MATH - 607/8 Contents Contents 1...

This preview shows document pages 1 - 5. Sign up to view the full document.

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