Proof of Stone-Wierstrass theorem

In summary, the conversation discusses the proof of the Stone-Wierstrass theorem, specifically the definition of a particular set and how to show that it is open. A possible solution is suggested, involving rewriting the set in a way that makes it clear that it is open.
  • #1
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Homework Statement



During the proof of the Stone-Wierstrass theorem we define a particular set and claim it is open. I can't figure out how to show it is open.
I wrote out the proof until that point to show how it is constructed, I apologize for the long windedness.

Assume that X is a compact hausdorf space[tex] A\subset C(X) [/tex]is an algebra that sperates points and has a continuosu function in it. then [tex]\overline{A}=C(X).[/tex]

Proof:

Since A separates points then for any [tex]a,b\in X[/tex] there exists [tex] h\in A [/tex] such that [tex] h(a)\neq h(b) [/tex]. define [tex] g(x)=\frac{h(x)-h(a)}{h(b)-h(a)}. [/tex] Then [tex] g(a)=0 [/tex]and [tex] g(b)=1[/tex].

for any function [tex] f\in A [/tex] define [tex] f_{\left(a,b\right)}(x)=\left(f(b)-f(a)\right)g(x)+f(a)\text{then }f_{\left(a,b\right)}(a)=f(a)\text{ and }f_{\left(a,b\right)}(b)=f(b) [/tex].

Set [tex]\epsilon>0 [/tex] and define [tex] U_{\left(a,b\right)}=\left\{ x\quad|f_{\left(a,b\right)}(x)<f(x)+\epsilon\right\} .[/tex]

My question is how do i gaurantee that [tex] U_{\left(a,b\right)} [/tex] is open? My professor took it for granted. I'm sure it has to do with the pre-image of f but i can't write it out.

Homework Equations





The Attempt at a Solution


 
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  • #2
With such a questions, you have to rewrite [tex]U_{(a,b)}[/tex] in such a way so that it is clear that it's open. In this case:

[tex]U_{(a,b)}=\{x~\vert~f_{(a,b)}(x)<f(x)+\epsilon\}=\{x~\vert~(f_{(a,b)}-f)(x)<\epsilon\}=(f_{(a,b)}-f)^{-1}(]-\infty,\epsilon[)[/tex].
 

1. What is the Stone-Weierstrass theorem?

The Stone-Weierstrass theorem is a fundamental result in mathematical analysis that states any continuous function on a compact interval can be uniformly approximated by a polynomial function.

2. Who discovered the Stone-Weierstrass theorem?

The theorem was jointly discovered by mathematicians Marshall Stone and Karl Weierstrass in the early 20th century.

3. What is the significance of the Stone-Weierstrass theorem?

The Stone-Weierstrass theorem is important because it provides a powerful tool for approximating complicated functions with simpler ones. It has applications in many areas of mathematics, including functional analysis, differential equations, and geometry.

4. Can the Stone-Weierstrass theorem be extended to higher dimensions?

Yes, the Stone-Weierstrass theorem can be extended to higher dimensions. In fact, there are several generalizations of the theorem, such as the Arzelà-Ascoli theorem and the Hahn-Banach theorem, which apply to more general spaces than just compact intervals.

5. Are there any limitations to the Stone-Weierstrass theorem?

While the Stone-Weierstrass theorem is a powerful and widely applicable result, it does have limitations. For example, it only applies to continuous functions and cannot be used to approximate discontinuous functions. Additionally, the theorem only guarantees uniform approximation, not pointwise convergence.

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