What is the Stone-Weierstrass Theorem?

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In summary, the Stone-Weierstrass theorem states that a subalgebra of a compact space which separates points is dense in the space.
  • #1
Oxymoron
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My version of the Stone-Weierstrass theorem is:

Let [itex]X[/itex] be a compact space, and suppose [itex]A[/itex] is a subalgebra of [itex]C(X,\mathbb{R})[/itex] which separates points of [itex]X[/itex] and contains the constant functions. Then [itex]A[/itex] is dense in [itex]C(X,\mathbb{R})[/itex].

I can see why the subalgebra would have to separate points (this is how A is dense, no?) but I don't understand HOW it separates points. I mean, isn't [itex]A[/itex] a subalgebra? Which means it is a subset of [itex]X[/itex] and has the same algebraic structure as [itex]X[/itex].

If this subset, [itex]A[/itex] were to separate points of [itex]X[/itex], then if I take any pair of distinct points [itex]x,y \in X[/itex] then there must exist a function [itex]f \in A[/itex] such that [itex]f(x) \neq f(y)[/itex].

Oh, I think I just answered this myself! So if the subalgebra did not separate points, then [itex]A[/itex] is never dense in [itex]X[/itex].

My gripe is, [itex]A[/itex] is a subalgebra of [itex]C(X,\mathbb{R})[/itex], what is [itex]C(X,\mathbb{R})[/itex]? And why must the subalgebra contain the constant function? What if it didn't contain the constant function?
 
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  • #2
Looks to me like you need to review some basic definitions! [itex]C(X,\mathbb{R})[/itex] is the set of all continuous functions from [itex]X[/itex] to [itex]\mathbb{R}[/itex]. As for " has the same algebraic structure as [itex]X[/itex]", X doesn't have any algebraic structure- it's just a compact topological space.
 
  • #3
Looks to me like you need to review some basic definitions!
It's always the definitions isn't it Halls? ;)
[itex]C(X,\mathbb{R})[/itex] is the set of all continuous functions from [itex]X[/itex] to [itex]\mathbb{R}[/itex]. As for " has the same algebraic structure as X", X doesn't have any algebraic structure- it's just a compact topological space.
That's what I thought!


Definition of a subalgebra:

A subalgebra of the set of all continuous functions from [itex]X[/itex] to it's underlying field [itex]\mathbb{F}[/itex] is a subspace [itex]A[/itex] such that for [itex]f,g \in A[/itex], [itex]fg \in A[/itex], that is, multiplication of elements of a subalgebra is closed.


Now if you have a compact space [itex]X[/itex], and suppose [itex]A[/itex] is a subalgebra (which means that multiplication defined
above is closed in [itex]A[/itex]) which separates points of [itex]X[/itex] and contains the constant functions. Then the subalgebra [itex]A[/itex] is dense in [itex]C(X,\mathbb{R})[/itex].
Im trying to understand why the part in red needs to be included in this theorem. Why does the subalgebra have to contain the constant functions?
 
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  • #4
Why does the subalgebra have to contain the constant functions?
Try considering special cases! What if X is a one-point set? A two-point set?
 
  • #5
why don't you try proving the polynomials are dense in the space of all continuous functions on an interval? maybe you will see what is needed for the proof.[and what if you consider the set of all those continuous functions vanishing at a given point?]
 

1. What is the Stone-Weierstrass Theorem?

The Stone-Weierstrass Theorem is a fundamental result in mathematical analysis that states that any continuous function on a compact interval can be uniformly approximated by polynomials. In other words, it shows that polynomials are dense in the space of continuous functions.

2. Who discovered the Stone-Weierstrass Theorem?

The Stone-Weierstrass Theorem was independently discovered by mathematicians Juliusz Schauder and Karl Weierstrass in the late 19th century. It is named after them and mathematician Marshall Stone, who also made significant contributions to the theorem.

3. What are the applications of the Stone-Weierstrass Theorem?

The Stone-Weierstrass Theorem has numerous applications in mathematics and other fields such as physics and engineering. It is used in approximation theory, functional analysis, and differential equations, among others. It also plays a key role in the proof of the fundamental theorem of algebra.

4. What are the conditions for the Stone-Weierstrass Theorem to hold?

The Stone-Weierstrass Theorem holds for any compact interval and any continuous function. However, it requires the set of functions over the interval to be closed under addition, scalar multiplication, and composition. This means that the set must be an algebra.

5. Are there any generalizations of the Stone-Weierstrass Theorem?

Yes, there are several generalizations of the Stone-Weierstrass Theorem that extend its applicability to more general spaces and functions. These include the Arzelà-Ascoli theorem, the Tietze extension theorem, and the Gelfand-Neumark theorem. Each of these theorems has its own specific conditions and applications.

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