For $V$ a finite dimensional vector space over a field $\mathbb{K}$, I have encountered the claim that $ \dim(\mathrm{Hom}(V,V)) = \dim(\mathrm{Hom}(V \times V, \mathbb{K})) $
where $\mathrm{Hom}(V,V)$ denote the vector spaces, respectively, of all linear maps from $V$ to $V$ and all bilinear maps from $V\times V$ to the ground field $\mathbb{K}$. I'm sure I'm overlooking something elementary, but I don't see this.
There is a theorem that, in general, for any finite-dimensional vector spaces $V$ and $W$ that $ \dim(\mathrm{Hom}(V,W)) = \dim(V)\dim(W) $
But, $\dim(V \times W) = \dim(V) + \dim(W)$ and therefore $ \dim(\mathrm{Hom}(V \times V, \mathbb{K})) = (\dim(V) + \dim(V))\cdot \dim(K) = 2\dim(V)\cdot 1 $ which is obviously not equal to $\dim(\mathrm{Hom}(V,V)) = \dim(V)\cdot\dim(V)$
Where is my mistake?