Proposition 1

— The kernel of $c_E : E \to E''$ is the closure of 0 in $E$. If $E$ is Hausdorff, $c_E$ is injective.

By construction, the kernel of $c_E$ is the intersection of the kernels of the continuous linear forms on $E$, i.e. the closure of $\{0\}$ in $E$ (II, p. 24, cor. 1).

When $E$ is Hausdorff, we identify $E$ with a subspace of $E''$, by means of the mapping $c_E$.

BIDUAL. REFLEXIVE SPACES

TVS IV.15

The strong topology on $E''$ is the $\mathscr{S}$-topology, where $\mathscr{S}$ is the family of all strongly bounded subsets of $E'$. Since every equicontinuous subset of $E'$ is strongly bounded (III, p. 22, prop. 9), the initial topology on $E$ is coarser than the topology obtained by taking the inverse image under $c_E$ of the strong topology on $E''$; it can be strictly coarser (IV, p. 52, exerc. 1). However: