Let $H$ be a separable Hilbert space and let $E:L^2([0,1],H)$ be the set of all continuous paths into $H$ which are continuous. How can we make $E$ into a separable Hilbert-space? I'm thinking the completion of the subset of $E$ for which the inner-product $$ <f,g> \triangleq \int_0^1 \langle f_t,g_t\rangle_H dt, $$ is finite, defines a complete inner product on the path space with respect to the above inner product, where $\langle x,y\rangle_H$ is the inner product on $H$.
But I'm uncertain if I made a mistake; have I overlooked something?
For example if $H=L^2(U;\mathbb{R})$, where $U$ is a Borel subset of $\mathbb{R}$, then is: $$ <f,g> \triangleq \int_0^1 \langle f_t,g_t\rangle_H dt = \int_0^1 \int_U f_t(u),g_t(u) du dt, $$ , $$