Proposition 16

(i) Let K, L be two fields, E a left vector space over K, F a left vector space over L and G a (K, L)-bimodule. Then the canonical $\mathbf{Z}$-homomorphism

(§ 4, no. 2, formula (7)) is injective; it is bijective when one of the vector spaces E, F is finite-dimensional.

(ii) Let $E_1, E_2, F_1, F_2$ be four vector spaces over a commutative field K; then the canonical K-homomorphism

(§ 4, no. 4, formula (21)) is injective; it is bijective if one of the ordered pairs $(E_1, E_2), (E_1, F_1), (E_2, F_2)$ consists of finite-dimensional spaces.

Assertion (i) is a particular case of § 4, no. 2, Proposition 2. Similarly the second assertion of (ii) is a particular case of § 4, no. 4, Proposition 4. Finally, to see that the homomorphism (28) is always injective, observe that $\operatorname{Hom}(E_i, F_i)$ is a vector subspace of $F_i^{E_i}$ ($i = 1, 2$) and that

is canonically identified with a vector subspace of the space $(F_1 \otimes F_2)^{E_1 \times E_2}$ (II, § 3, no. 1, Proposition 1); when these identifications are made and also the left hand side of (28) is identified with a subspace of $F_1^{E_1} \otimes F_2^{E_2}$ (Proposition 14), the canonical mapping (28) becomes the restriction to this subspace of the canonical mapping (26) and it has been seen (Proposition 15) that the latter is injective.