Hilbert space to bound a linear operator from span of monomials into l2

Question

I am looking for a Hilbertspace $H$ which contains the monomials and such that the linear operator

$\begin{align*}U \colon span\{1,x^1,x^2,...\} &\to l^2_F(\mathbb{N,\mathbb{C}})\\ x^n &\mapsto e_n \end{align*}$

is bounded in both directions. Here, $e_n$ denotes the n-th canonical basis vector of $l^2_F$, the space of complex sequences with finite support endowed with the $l^2$-norm.

It's fairly easy to see that the space $L^2(0,1)$ does not deliver boundedness. The next thing that came to my mind was the Sobolev space $H^1(0,1)$. Showing boundedness would result in proving the inequality

$\sum_{n=1}^{N} \lvert c_n \rvert^2 \leq const \lVert p \rVert_{H^1(0,1)}^2$

for a polynomial $p$ of degree $N$ and its coefficients $c_n$.but I don't know how to bound the sum of the squares of the coefficients of $p$ in $H^1$-norm (or in any other norm, OR if it is even possible).

Thanks in advance

Answer 1

You can always define an inner product on $\operatorname{span}\{1,x,x^2,\ldots\}$ where the monomials form an orthonormal set. Then your map becomes automatically an isometry.