Center of general linear group Given a (not necessarily finite dimensional) vector space $V$ prove that the center of $\operatorname{GL}(V)$ is the set of all scalar transformations (i.e all transformations of the form $a\operatorname{Id}$)? 
I know how to prove this for general linear group of degree $n$, please help me solve this for the case of a general linear map.

Let $T$ in the center. For any $L$ we have $T\circ L= L\circ T$, that is 
$$T(Lx) = L(Tx)$$ for all $L$ and all $x \in V$. 
Let $x$ in $V$. There exists  $L$ linear map so that the subspace $\{y \ |  \ Ly = y\}$ equals $\mathbb{F} \cdot x$ ( use a basis starting from $x$).
We get 
$L(Tx) = T(Lx) = Tx$ and so $Tx \in \mathbb{F} \cdot x$. 
So, for any $x\in V$ we have $T(x)$ proportional to $x$. It's easy now.
