Example 4

When $F = \mathbf{K}$, the space $\mathscr{L}(E; F)$ is the dual $E'$ of E. We denote by $E'_\mathfrak{S}$, $E'_s$ etc. the space $\mathscr{L}_{\mathfrak{S}}(E; \mathbf{K})$, $\mathscr{L}_s(E; \mathbf{K})$ etc. The space $E'_s$ (resp. $E'_b$) is called the weak dual (resp. strong dual) of E. A bounded subset of $E'_s$ (resp. $E'_b$) is said to be weakly (resp. strongly) bounded. We observe that the weak topology on $E'$ is none other than $\sigma(E', E)$ (II, p. 42).

When $E = F$, we denote by $\mathscr{L}(E)$, $\mathscr{L}_{\mathfrak{S}}(E)$ etc. the space $\mathscr{L}(E; F)$, $\mathscr{L}_{\mathfrak{S}}(E; F)$ etc.

Let $p$ be a continuous semi-norm on F and M a bounded subset of E. Let

It is immediate that $p_M$ is a semi-norm on $\mathscr{L}(E; F)$ and that if $\Gamma$ is a fundamental system of semi-norms on F, the family of semi-norms $p_M$, where $p$ ranges over $\Gamma$ and M ranges over a base for the bornology generated by $\mathfrak{S}$, is a fundamental system of semi-norms of $\mathscr{L}_{\mathfrak{S}}(E; F)$.

In particular, if E and F are semi-normed spaces, and if $p$ (resp. $q$) denotes the semi-norm of E (resp. F), then the topology of bounded convergence on $\mathscr{L}(E; F)$ is defined by the semi-norm

(cf. GT, X, § 3, No. 2). When we consider $\mathscr{L}_b(E; F)$ as a semi-normed space, we shall always, unless the contrary is expressly stated, mean the semi-norm (2). If F is a normed space, the semi-norm (2) is a norm.