In the context of vertex operator algebras, if $V$ is a vector space, how can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\operatorname{End}V)[[x,x^{-1}]]$?
The notation $V((x))$ is the space of formal Laurent series truncated from below.
A line in a text by Lepowski says to make this natural identification, but it's not clear to me what this is supposed to be.
If you have a linear map $F:V\to V((x))$, then the coefficient of $x^i$ in $F(v)$ depends linearly on $v$. Hence you can write $F(v)=\sum_{i\in\mathbb Z} f_i(v)x^i$ where each $f_i:V\to V$ is linear. (In addition, you know that for each $v\in V$, there is some $i_0$ such that $f_i(v)=0$ for $i<i_0$, but this does not play a role here.) Now you simply associate to $F$ the formal Laurent series $\sum_{i\in\mathbb Z}f_ix^i$.