Letting $\mathbb{C}_0[X]$ be the space of complex polynomials without constant term and $(\kappa_n)$ a sequence of real numbers (To be precise, the $\kappa$'s are the free cumulants of a compactly supported distribution on $\mathbb{R}$) such that $$\langle X^n,X^m \rangle = \kappa_{n+m}$$ extends by linearity to an inner product on $\mathbb{C}_0[X]$, i.e
$$\langle \sum\alpha_nX^n,\sum\beta_nX^n \rangle = \sum_j \sum_k\alpha_j\bar{\beta_k}\kappa_{j+k}$$
and we also assume there is $c>0$ such that $|\kappa_n|<c^n$ for all $n$.
I want to show that if $(Y_n)$ is a Cauchy-sequence under the induced norm, then $(XY_n)$ is also a Cauchy-sequence. The motivation is that I'm considering the Hilbert space completion of this space, and I want to extend multiplication by $X$ to the entire Hilbert space, not just the embedding of $\mathbb{C}_0[X]$.
Writing $Y_n = \sum_{j=1}^{K_{n,m}}\alpha_{j,n}X^j$ and similarly for $Y_m=\sum_{j=1}^{K_{n,m}}\alpha_{j,m}X^j$ such that both sums have the same number of terms, adding $0$ coefficients whenever needed, the assumption is that $$\|Y_n-Y_m \|^2 = \sum_{i=1}^{K_{n,m}}\sum_{j=1}^{K_{n,m}}(\alpha_{i,n}-\alpha_{i,m})\overline{(\alpha_{j,n}-\alpha_{j,m})}\kappa_{j+k} \to 0$$ and I want to show that
$$\|X(Y_n-Y_m) \|^2 = \sum_{i=1}^{K_{n,m}}\sum_{j=1}^{K_{n,m}}(\alpha_{i,n}-\alpha_{i,m})\overline{(\alpha_{j,n}-\alpha_{j,m})}\kappa_{(j+1)+(k+1)} \to 0 $$ but I think the two expressions are hard to compare or to manipulate in any meaningful way. Any help is appreciated.