$\mathcal{L}(V,W)$ is infinite dimensional when $V$ is finite dimensional and $W$ is infinite dimensional.

I did not read you proof, but to show that $L(V, W)$ have infinite dimension, you can argue that the maps of the form $f_{i,j}: V \to W$ s.t

$$f_{i,j}(v_k) = \begin{cases} w_j & k = i\\ 0 & k\not = i\\ \end{cases},$$ - where $\{v_i\}$ and $\{w_j\}$ forms a basis for $V$ and $W$, respectively - generates a subspace of $L(V, W)$.Therefore, since there are infinitely many such maps, and the space generated by such maps is a subspace of $L(V,W)$, we must have $\dim L(V, W)$ has to be infinite.