Stone Weierstrass application?

[nalgas]

Homework Statement

I want to prove that the span of $\{x^{2n}:n \geq0\}$ is dense in $C([0,1])$.

Furthermore, that the closure of the span of $\{x^{2n+1}:n \geq0\}$ is $\{f \in C([0,1]):f(0) = 0\}$.

Homework Equations

Is my solution correct?

Now I do not know how to tackle the second part. Any help?

Thank you for helping the community.

The Attempt at a Solution

My work:

I guess I can use the Stone-Weierstrass theorem.

Define $A =$ span of $\{x^{2n}:n \geq0\}$.
I need to prove A is an algebra of real continuous functions on a compact set I = [0,1]. The set I is compact since it is continuous and bounded.

A is an algebra since if, for example I multiply $fg = x^2x^4 = x^8 \in A$.

Next, need to show A separates points on I. So, points $x_1,x_2 \in [0,1] \Rightarrow f(x_1) \neq f(x_2), f(x_1), f(x_2) \in \{x^{2n}:n \geq0\}$.

Finally, A vanishes at no point of I, i.e. $\forall x \in [0,1]$ elements of $\{x^{2n}:n \geq0\}$ are $\neq0$.

 

[wabbit]

Yes, you can certainly do that, it will work fine. Or, if you're lazy, you can just recycle the fact that polynomials are dense (presumably you've already proved that since you're using Weierstrass), by using a simple transformation (homeomorphism of [0,1] with itself), do you see which one ?