Corollary 3

Let S be a set and V a vector subspace of the right vector K-space $K_d^S$ of mappings of S into K. In order that $\dim V \geq p$ (where p is an integer), it is necessary and sufficient that there exist p elements $s_i$ of S and p elements $f_i$ of V $(1 \leq i \leq p)$ such that $f_i(s_j) = \delta_{ij}$.

The space $K_d^S$ is canonically identified with the dual of $E = K_d^{(S)}$ and $f(s) = \langle e_s, f \rangle$ for $s \in S$ and $f \in K_s^S$, $(e_s)_{s \in S}$ being the canonical basis of E. Corollary 2 then shows that the condition is sufficient. Conversely, suppose that $\dim V \geq p$, so that there exists a subspace $G'$ of V of dimension $p$; let F be the orthogonal of $G'$ in E, so that $\dim(E/F) = p$. It follows from no. 1, Theorem 2 that there exist $p$ elements $s_i \in S$ such that the $e_{s_i}$ $(1 \leq i \leq p)$ form a basis of a supplement F of E (applying Theorem 2 (no. 1) to a free subset consisting of a basis of E and the generating system the union of this free subset and the canonical basis of E); we then take the $f_i$ to be the elements of a basis of G' dual to the basis of E/F consisting of the classes of the $e_{s_i}$ mod. F.