We know that there is a natural isomorphsm between $$V^*\otimes W \text{ and } Hom(V,W)$$ whenever either $V$ or $W$ is finite dimensional. (We also know that there always exists a linear map from $$V^*\otimes W \rightarrow Hom(V,W)$$ regardless of their dimension.
I am looking for an explicit example why such isomorphism does not exist if both are infinite dimensional. I am not sure how important this fact is, but I am merely asking out of curiosity.
Notice that the linear map of $γ\colon V^* \otimes W → \operatorname{Hom}(V,W)$ comes from the bilinear map $$V^* × W → \operatorname{Hom}(V,W),\quad (φ,w) ↦ (·w)∘φ.$$ Now for any $(φ,w) ∈ V^* × W$, $\dim \operatorname{img} ((·w) ∘ φ) ≤ 1$. Any element $x$ in $V^* \otimes W$ is a finite linear combination of the generating elements $φ \otimes w$, so the dimension of the image of $γ(x)$ must be finite, too.
So you don’t even hit the identity $ℚ^ℕ → ℚ^ℕ$ with $γ$ if $V = W = ℚ^ℕ$.
