Definition 13.12.2 (Bi-Equicontinuous Convergence).label Let $E, F$ be locally convex spaces over $K \in \RC$ and
\[\sigma = \bracsn{S \times T| S \subset E^* \text{ equipcontinuous}, T \subset F^* \text{ equicontinuous}}\]
be the product of all equicontinuous subsets of $E^{*}$ and $F^{*}$, then the $\sigma$-topology on $L^{2}(E, F; K)$ is the topology of bi-equicontinuous convergence on $L^{2}(E, F; K)$. Under this topology, $L^{2}(E, F; K)$ is a locally convex space.
Proof. By Proposition 13.10.1, the $\sigma$-topology is a vector space topology.$\square$
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