For each $n\in\mathbb{N}$ , can $x^n$ be uniformly approximated by the linear combination of $\left(x^{k^2}\right)_{k\in\mathbb{N}}$ ?
In order to facilitate a solution, we might as well try to approximate $x^n$ only on $[0,1]$ first.
My Attempt
At the beginning I wanted to prove that every continuous function can be uniformly approximated by the linear combination of $(x^{k^2})_{k\in\mathbb{N}}$ on $[0,1]$. But from a theorem of Müntz, I found this idea wrong.
Considering that $\displaystyle\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$, $\Pi=\text{span}\{x^{k^2}:k\in\mathbb{N}\}$ isn't a dense subset of $C[0,1]$ .
On the other hand, $\tilde\Pi=\text{span}\{x^{k}:k\in\mathbb{N}\}$ is a dense subset of $C[0,1]$. And if $x^n$ can be approximated, it yields that every polynomial can be approximated by the linear combination of $(x^{k^2})_{k\in\mathbb{N}}$.
Does this imply that not all the $x^n$ can be uniformly approximated by the linear combination of $(x^{k^2})_{k\in\mathbb{N}}$? But actually I was asked to prove that it's right.
Any helps? Thanks in advance!