Irreducible Representations and Maschke's Theorem 
$\mathscr L(V,W)^G = \mathscr L(V_1,W)^G \oplus...\oplus\mathscr
 L(V_k,W)^G$ and for each irreducible represtation of G on a space W,
   the number of $j\in (1,...,k)$ for which $V_j \cong W$ is dim$\mathscr
 L(V,W)^G$

I've proven that for each representation of a finite group G on a space V, we have a direct sum decomposition of V into stable irreducible subspaces, $V = V_1 \oplus ... \oplus V_k$.  And I know that dim $\mathscr L (V,W)=1$ if $V \cong W$ and $0$ otherwise.  I'm not sure how to go about this though.  
EDIT: To clarify, $\mathscr L (V,W)^G$ means that $gTv=Tgv$ for all $g\in G, v\in V$
 A: Allow me to write $\text{Hom}_G(V,W)$ in lieu of $\mathscr{L}(V,W)^G$ and --script letters weird me out. Also, let $\text{Hom}(V,W)$ without the $G$ denote vector space maps. You have an obvious vector space isomorphism
$$\displaystyle f:\bigoplus_{j=1}^{k}\text{Hom}(V_j,W)\to \text{Hom}(V,W)$$
By merely taking a $k$-tuple of transformations $(T_1,\cdots,T_k)$ to the transformation $\displaystyle \bigoplus_{j=1}^{k}T_j$ action on $T$ by action on each of coordinates $V\cong V_1\oplus\cdots\oplus V_k$. Since $T$ clearly commutes with the $G$-action if and only if $f(T)$ we actually see that this $f$ restricts to an isomorphism
$$\displaystyle f_G:\bigoplus_{j=1}^{k}\text{Hom}_G(V_j,W)\to \text{Hom}_G(V,W)$$
Then, we see that 
$$\begin{aligned}\dim\text{Hom}_G(V,W) &= \dim\left(\bigoplus_{j=1}^{k}\text{Hom}_G(V_j,W)\right)\\ &= \sum_{j=1}^{n}\dim\text{Hom}_G(V_j,W)\\ &= \sum_{j:V_j\cong W}\dim\text{Hom}_G(V,W)+\sum_{j:V_j\not\cong W}\dim\text{Hom}_G(V_j,W)\\ &= \sum_{j:V_j\cong W}1+\sum_{j:V_j\not\cong W}0\\ &= \#\{j:V_j\cong W\}\end{aligned}$$
