Corollary

Let $F$ be a left vector space over $K$; for every set $X$, the left vector space $K_d^X \otimes_K F$ is identified with the subspace of the space $F^X$ of all mappings of $X$ into $F$, consisting of the mappings $u$ such that $u(X)$ is of finite rank in $F$.

If $(f_\lambda)$ is a basis of $F$, the element $\sum_{\lambda \in L} v_\lambda \otimes f_\lambda$ of $K_d^X \otimes_K F$ is identified under (26) with the mapping $x \mapsto \sum_\lambda v_\lambda(x) f_\lambda$. As $v_\lambda = 0$ except for the indices $\lambda$ belonging to a finite subset $H$ of $L$, the image of $X$ under the above mapping is contained in the finite-dimensional subspace of $F$ generated by the $f_\lambda$ of indices $\lambda \in H$. Conversely, let $u : X \to F$ be a mapping such that $u(X)$ is contained in a finite-dimensional subspace $G$ of $F$ and let $(b_i)_{1 \leq i \leq n}$ be a basis of $G$. For all $x \in X$, we may write $u(x) = \sum_{i=1}^n v_i(x) b_i$, where the $v_i(x)$ are well determined elements of $K$; thus $n$ mappings $v_i : X \to K$ are defined and clearly $u$ is then identified with the element $\sum_{i=1}^n v_i \otimes b_i$.

Similarly, for a right vector space $E$ over $K$ and a set $Y$, $E \otimes_K K_s^Y$ is identified with a subspace of the space $E^Y$, consisting of the mappings $v : Y \to E$ such that $v(Y)$ is of finite rank. More particularly, for every field $K$, $K_d^X \otimes_K K_s^Y$ is identified with a subspace of the space $K^{X \times Y}$ of mappings of $X \times Y$ into $K$ ($K$ being considered as a $(K, K)$-bimodule); an element $\sum_i u_i \otimes v_i$, where $u_i$ is a mapping of $X$ into $K$ and $v_i$ a mapping of $Y$ into $K$, is identified with the mapping $(x, y) \mapsto \sum_i u_i(x) v_i(y)$ of $X \times Y$ into $K$.