Definition 13.12.5 (Integral Bilinear Form).label Let $E, F$ be locally convex spaces over $K \in \RC$ and $\lambda \in L^{2}(E, F; K)$ be a bilinear form, then $\lambda$ is integral if there exists equicontinuous subsets $S \subset E^{*}$ and $T \subset F^{*}$, and a Radon measure $\mu \in M_{R}(S \times T; K)$ such that
\[\lambda(x, y) = \int_{S \times T}\dpn{x, \phi}{E}\dpn{y, \psi}{F}\mu(d\phi, d\psi)\]
for all $x, y \in E$.
The set $I(E, F)$ is the space of integral bilinear forms on $E$ and $F$.
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