To proof this statement I will show that:
$\quad dim(\mathcal{V}) \leq dim(\mathcal{W})$|$\quad dim(\mathcal{W}) \leq dim(\mathcal{V})$
(1) $dim(\mathcal{V}) \leq dim(\mathcal{W})$
(2) $dim(\mathcal{W}) \leq dim(\mathcal{V})$
From this follows that dim(\mathcal{V}) = dim(\mathcal{W})$.
Proof of (1):