Exercise 27, § 7

(a) Under the hypotheses of Proposition 16 (i) (no. 7), for the canonical homomorphism (27) (no. 7) to be bijective, it is necessary and sufficient that $E$ or $F$ be finite-dimensional (note that the image under (27) (no. 7) of an element of $\operatorname{Hom}_K(E, G) \otimes_L F$ maps $E$ to a subspace of $G \otimes_L F$ contained in a subspace of the form $G \otimes_L F'$, where $F'$ is a finite-dimensional subspace of $F$).

(b) Under the hypotheses of Proposition 16 (ii) (no. 7), for the canonical homomorphism (28) (no. 7) to be bijective, it is necessary and sufficient that one of the ordered pairs $(E_1, E_2), (E_1, F_1), (E_2, F_2)$ consist of finite-dimensional vector spaces (use (a) and Exercise 7 of § 4).