Exercise 26, § 7

(a) Let $E, F$ be two left vector spaces over a field $K$; for the canonical homomorphism $E^* \otimes_K F \to \operatorname{Hom}_K(E, F)$ to be bijective, it is necessary and sufficient that one of the spaces $E, F$ be finite-dimensional.

(b) Let $E$ be a right vector space and $F$ a left vector space over a field $K$. For the canonical homomorphism $E \otimes_K F \to \operatorname{Hom}_K(E^*, F)$ to be bijective, it is necessary and sufficient that $E$ be finite-dimensional (use Exercise 3 (d)).