- #1
nalgas
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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 $\neq0$.