Remark 1

If $E$ is a locally convex space and $N$ is the closure of $\{0\}$ in $E$, so that $E_1 = E/N$ is the Hausdorff locally convex space associated with $E$, we know that the duals $E'$ and $E'_1$ are identical; for a function $f$ to belong to $\widetilde{\mathscr{K}}(X; E)$, it is necessary and sufficient that $f_1 = \pi \circ f$ (where $\pi : E \to E_1$ is the canonical homomorphism) belong to $\widetilde{\mathscr{K}}(X; E_1)$, in which case $\int f d\mu = \int f_1 d\mu$. We may therefore limit ourselves to considering only Hausdorff locally convex spaces.