Exercise 25, § 7

(a) Let $K$ be a field and $L, M$ two sets; there are canonical inclusions $(K_s^L)^{(M)} \subset (K_s^{(M)})^L \subset K_s^{L \times M}$. Show that for two of these vector spaces to be equal, it is necessary and sufficient that one of the sets $L, M$ be finite.

(b) Let $(E_\lambda)_{\lambda \in L}$ be a family of right vector spaces over $K$ and $(F_\mu)_{\mu \in M}$ a family of left vector spaces over $K$. For the canonical mapping (26) (no. 7) to be bijective, it is necessary and sufficient that one of the vector spaces $\prod_{\lambda \in L} E_\lambda, \prod_{\mu \in M} F_\mu$ be finite-dimensional (use (a)).