Date of Award

1-1-2013

Document Type

Open Access Thesis

Department

Mathematics

First Advisor

Maria Girardi

Abstract

In this thesis we will consider the work began by Weierstrass in 1855 and several generalization of his approximation theorem since. Weierstrass began by proving the density of algebraic polynomials in the space of continuous real-valued functions on a finite interval in the uniform norm. His theorem has been generalized to an arbitrary compact Hausdorff space and the approximation with elements from more general algebras of continuous real-valued functions. We will consider proofs that use brute force and proofs based on convolutions and approximate identities, trudge through probability and the use of the Bernstein polynomials, and become intimately close to what is meant by an algebra. We will also see what inspired Weierstrass and Stone throughout their life as we take a sneak peak into their Biographies.

Included in

Mathematics Commons

Share

COinS