Remark

(1) Let K be a commutative field, E, F two vector spaces over K, (a_\lambda) a basis of E and (b_\mu) a basis of F; then (a_\lambda \otimes b_\mu) is a basis of the vector K-space E \otimes_K F (II, § 3, no. 7, Corollary 2 to Proposition 2) and therefore

(2) Let K be a commutative field, E_1, E_2, F_1, F_2 four vector spaces over K and u:E_1 \to F_1, v:E_2 \to F_2 two linear mappings; then It is immediate that $(u \otimes v)(E_1 \otimes E_2)$ is the canonical image of $u(E_1) \otimes v(E_2)$ in $F_1 \otimes F_2$ and hence (Proposition 14) is isomorphic to $u(E_1) \otimes v(E_2)$; the conclusion then follows from (30). (3) Under the same hypotheses as in Remark 1, If E is isomorphic to $K^{(I)}$, $\operatorname{Hom}(E, F)$ is isomorphic to $(\operatorname{Hom}(K, F))^I$ (§ 1, no. 6, Corollary 1 to Proposition 6) and hence to $F^I$ (§ 1, no. 14); as $F^{(I)}$ is a subspace of $F^I$ and $\dim(F^{(I)}) = \operatorname{Card}(I) \cdot \dim F = \dim E \cdot \dim F$ (no. 2, Proposition 2), the inequality (32) follows from no. 3, Corollary 4 to Proposition 4. The same argument shows that the two sides of (32) are equal when $\dim E$ is finite (cf. § 10, nos. 3 and 4).