I am reading this book about vertex Operator Algebra; I have difficulty understanding and verifying this statement in page 26:-
Consider $Hom(V, V ((x)))$ as a natural subspace of $(End\, V)[[x , x^{-1}]]$.
Where $V$ and $V((x))$ are vector space (the latter is space of truncatedformal Laurent series). and $V[[x,x^{-1}]]$ is the vector space of (doubly infinite) formal Laurent series in X with coefficients in V. any explanation or making sense of this statement appreciated.
First, it should be clear that $V((x))$ is a subspace of $V[[x, x^{-1}]]$: The former only allows finitely many negative powers of $x$, while the latter allows infinitely many positive or negative powers of $x$.
Now let's consider an element $\phi \in Hom(V, V((x)))$. Then $\phi: V \to V((x))$, and we can denote the image of an element $v\in V$ by $\phi(v) = \sum_{n=N}^{\infty}v_nx^n$, where $v_n\in V$ for each $n\in\mathbb{Z}$. So for each $n\in \mathbb{Z}$, we have an element, $v_n\in V$, which is the coefficient of $x^n$ in $\phi(v)$. Denote the mapping from $v\mapsto v_n$ by $\phi_n$, and observe that $\phi_n\in End(V)$ for every $n\in\mathbb{Z}$. Therefore, $\phi$ defines an element of $End(V)[[x, x^{-1}]]$, given explicitly by $\phi = \sum_{n\in\mathbb{Z}}\phi_n x^n$.
It is often helpful to keep in mind that we usually don't think of formal power series (or formal Laurent series) as functions of $x$: There is no topology or convergence here. Rather, for the purposes of understanding these spaces, think of $\phi$ as a generating function for a family of endomorphisms of $V$, and consider the Laurent series notation merely a convenient way of indexing these endomorphisms.