Theorem 6

For every vector space E over a field K, the canonical mapping $c_E : E \to E^{**}$ (§ 2, no. 7) is injective; for it to be bijective, it is necessary and sufficient that E be finite-dimensional.

The first assertion and the fact that if E is finite-dimensional $c_E$ is bijective are special cases of § 2, no. 7, Proposition 14. Suppose that E is infinite-dimensional, so that we may assume that $E = K_s^{(L)}$, where L is an infinite set and therefore $E^* = K_d^L$. Let $(e_\lambda)_{\lambda \in L}$ be the canonical basis of E and let $(e_\lambda^*)_{\lambda \in L}$ be the corresponding family of coordinate forms in $E^*$ (§ 2, no. 6); the vector subspace of $E^*$ generated by the $e_\lambda^*$ is just the direct sum $F' = K_d^{(L)}$ and the hypothesis that L is infinite implies that $F' \neq E^*$. Then there exists a hyperplane $H'$ of $E^*$ containing $F'$ (no. 3, Proposition 8) and, as $E^*/H'$ is non-zero, its dual is also non-zero (Corollary to Theorem 4), which is identified with the orthogonal $H''$ of $H'$ in $E^{**}$ (§ 2, no. 6, Corollary to Proposition 9). But $H'' \cap c_E(E)$ is contained in the image under $c_E$ of the orthogonal of $F'$ in E, which is by definition 0; hence $c_E(E) = E^{**}$ is impossible.

E will usually be identified with the subspace of $E^{**}$ the image of $c_E$.

Let E, F be two vector spaces over a field K and $u : E \to F$ a linear mapping. We shall define canonical isomorphisms:

(1) Of the dual of Im(u) = u(E) onto Im($^t u$) = $^t u(F^*)$.

(2) Of the dual of Ker(u) = $^{-1}u(0)$ onto Coker($^t u$) = $E^*/^t u(F^*)$.

(3) Of the dual of Coker(u) = F/u(E) onto Ker($^t u$) = $^t u^{-1}(0)$.

We write I = Im(u), N = Ker(u), C = Coker(u); from the exact sequences

(14) $0 \to N \to E \xrightarrow{p} I \to 0,\quad 0 \to I \xrightarrow{j} F \to C \to 0$

we derive, by transposition (Corollary to Theorem 5), the exact sequences

(15)

Moreover, as $u = j \circ p,\ ^t u = ^t p \circ ^t j$; the exact sequences (15) thus define canonical isomorphisms of $C^*$ onto Ker($^t u$), of $I^*$ onto Im($^t u$) and of $N^*$ onto Coker($^t u$), since $^t p$ is injective and $^t j$ surjective. To be precise, let $y \in \mathrm{Im}(u)$, $z \in \mathrm{Ker}(u)$, $t \in \mathrm{Coker}(u)$, $y' \subset \mathrm{Im}(^t u)$, $z' \in \mathrm{Coker}(^t u)$, $t' \in \mathrm{Ker}(^t u)$; when $y'$, $z'$, $t'$ are canonically identified with linear forms on $\mathrm{Im}(u)$, $\mathrm{Ker}(u)$ and $\mathrm{Coker}(u)$ respectively, then

(16) $\langle y, y' \rangle = \langle x, y' \rangle$ for all $x \in E$ such that $u(x) = y$;

(17) $\langle z, z' \rangle = \langle z, x^* \rangle$ for all $x^* \in E^*$ whose class mod. $^t u(F)$ is equal to $z'$;

(18) $\langle t, t' \rangle = \langle s, t' \rangle$ for all $s \in F$ whose class mod. $u(E)$ is equal to $t$.

In particular we derive from these results: