Weak time continuity and energy spaces

The weak-continuity clause is the continuity of all scalar coordinate maps $t\mapsto (u(t),h{)}_H$ and therefore defines a property of an actual map; it makes no assertion of norm continuity. The Bochner conditions, by contrast, are properties of almost-everywhere equivalence classes.

If $u,v\in C_w([0,T];H)$ agree almost everywhere, then for each fixed $h\in H$ the continuous scalar function $t\mapsto (u(t)-v(t),h{)}_H$ vanishes almost everywhere. A continuous scalar function that is nonzero at any point is nonzero on a relative interval of positive measure, including at an endpoint, so this function vanishes everywhere. Taking $h=u(t)-v(t)$ gives $u(t)=v(t)$ for every $t$. Thus a weakly continuous representative, if it exists for a Bochner class, is unique.

Under the Countable Choice convention in [F1], the periodic targets are separable. Consider finite vector Fourier arrays with coefficients in $\mathbb{Q}+i\mathbb{Q}$, support in $\{\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le N\}$ for an integer $N$, and conjugate symmetry; their associated real polynomials form a countable family. For $H$ and $V$, apply $P_k$ to each coefficient. This preserves conjugate symmetry and gives finite solenoidal arrays; $P_0=I$ retains the mean mode. Given a target field, first truncate its coefficient series in its target norm, then approximate the finitely many coefficients by rational real and imaginary coordinates with the opposite modes paired by conjugation. Since $P_k$ is a contraction and fixes the target solenoidal coefficients, the projected rational approximants converge in the $H$ or $V$ norm. For the real $H^{-r}$ space, omit the projections: its definition as the weighted completion of finite conjugate-symmetric arrays gives the same density argument. Complex targets use the corresponding unpaired rational complex arrays. Thus every periodic target named here is separable; [F2] supplies the bounded-domain targets under its own Axiom of Choice hypothesis. Finally, $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p$ is exactly the conjunction of the Bochner-class conditions over finite intervals. No stronger assumption on the abstract spaces in the Definition is needed.