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Stone Weierstrass application?

  1. Mar 10, 2015 #1
    1. The problem statement, all variables and given/known data

    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 $\{f \in C([0,1]):f(0) = 0\}$.


    2. Relevant equations

    Is my solution correct?

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

    Thank you for helping the community.



    3. 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 $\neq0$.
     
  2. jcsd
  3. Mar 11, 2015 #2

    wabbit

    User Avatar
    Gold Member

    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 ?
     
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