If $V$ is a vector space over the field $F$, the general linear group of $V$, written $GL(V)$, is the set of all bijective linear transformations $V\to V$, together with functional composition as group operation.
My question: Let $V$ and $W$ be infinite dimensional vector spaces over a field $F$. Is the following statement true?
$GL(V)\cong GL(W)$ if and only if $\dim(V)=\dim(W)$.
Please provide references. Thanks in advance.