Intro to Analysis in-class_Part_60

# Intro to Analysis in-class_Part_60 - 7.5 Approximation by...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7.5 Approximation by polynomials 125 Proof. By Weierstrass’ Thm, there is a sequence { g n } of real polynomials converging uniformly to | x | . Then let f n ( x ) := g n ( x )- g n (0) . 7.5.2 The Stone-Weierstrass Theorem We will isolate the properties of the polynomials which makes such an approximation possible. Definition 7.5.13. A family of functions A defined on a set X is an algebra iff (i) f + g ∈ A , (ii) fg ∈ A , and cf ∈ A whenever f,g ∈ A and c is a constant. (If A is an algebra of complex functions, then c ∈ C .) Definition 7.5.14. An algebra A is uniformly closed (or closed in the topology of uniform convergence) iff f n unif----→ f,f n ∈ A = ⇒ f ∈ A . The uniform closure of A is the set of all functions which are limits of uniformly convergent sequences of elements of A . Example 7.5.1. The polynomials on [ a,b ] form an algebra of functions. Weierstrass’ Thm states that C [ a,b ] is the uniform closure of the polynomials on [ a,b ]....
View Full Document

## This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

### Page1 / 2

Intro to Analysis in-class_Part_60 - 7.5 Approximation by...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online