Bochner L2 on a finite interval is a separable Hilbert space

Pointwise Cauchy–Schwarz and scalar Cauchy–Schwarz show that the displayed pairing is integrable and bounded by $\mathrm{\parallel }u{\mathrm{\parallel }}_Z\mathrm{\parallel }v{\mathrm{\parallel }}_Z$. It is bilinear and symmetric; if $\int \mathrm{\parallel }u{\mathrm{\parallel }}_H^2=0$, then $\mathrm{\parallel }u(t){\mathrm{\parallel }}_H=0$ almost everywhere, so the equivalence class is zero. Thus it is a real inner product.

Let $(u_n)$ be Cauchy in $Z$. By [F4], choose a subsequence $(u_{n_k})$ with $\mathrm{\parallel }{u}_{n_{k+1}}-u_{n_k}{\mathrm{\parallel }}_Z\le 2^{-k}$. Use [F4] to choose strongly measurable representatives and modify them to zero on their exceptional null sets. Put $g_m(t)={\sum }_{k=1}^m\mathrm{\parallel }{u}_{n_{k+1}}(t)-u_{n_k}(t){\mathrm{\parallel }}_H$. Minkowski gives $\mathrm{\parallel }{g}_m{\mathrm{\parallel }}_2\le {\sum }_{k\ge 1}{2}^{-k}=1$; monotone convergence then gives $g={\sum }_k\mathrm{\parallel }{u}_{n_{k+1}}-u_{n_k}{\mathrm{\parallel }}_H\in L^2$. Hence $g<\mathrm{\infty }$ almost everywhere, and completeness of $H$ gives a pointwise limit $u(t)$ there; put $u=0$ on the null set.

The limit $u$ is strongly measurable: the chosen representatives are strongly measurable and converge pointwise off a measurable null set, so their simple approximations give a strongly measurable limit. The tail estimate $\mathrm{\parallel }u-u_{n_m}{\mathrm{\parallel }}_H\le {\sum }_{k\ge m}\mathrm{\parallel }{u}_{n_{k+1}}-u_{n_k}{\mathrm{\parallel }}_H$ and [F2] show $\mathrm{\parallel }u-u_{n_m}{\mathrm{\parallel }}_Z\le {\sum }_{k\ge m}{2}^{-k}\to 0$. Thus $u\in Z$ and the subsequence converges in norm. Since the original sequence is Cauchy, the full sequence converges to the same class; $Z$ is complete.

Fix the given countable dense set $D\subset H$. From strong measurability choose finite-valued simple functions $s_n$ converging pointwise almost everywhere to a representative $u$. Replace $s_n$ by $t_n=s_n{\mathbf{1}}_{\{\mathrm{\parallel }{s}_n{\mathrm{\parallel }}_H\le 2\mathrm{\parallel }u{\mathrm{\parallel }}_H\}}$; then $t_n\to u$ pointwise and $\mathrm{\parallel }u-t_n{\mathrm{\parallel }}_H^2\le 9\mathrm{\parallel }u{\mathrm{\parallel }}_H^2$, so dominated convergence makes finite-valued simple functions dense in $Z$. Let $\mathcal{A}$ be the algebra generated by rational intervals restricted to $(0,T)$. It is countable and generates the Borel sigma-algebra. To verify approximation by this particular algebra, let $\mathcal{M}$ consist of Borel sets approximable in symmetric-difference measure by members of $\mathcal{A}$. It contains $\mathcal{A}$, is closed under complements because symmetric differences are preserved by complements, and is closed under finite unions by subadditivity. For a countable union, first approximate it by a finite subunion: the measure of the omitted part tends to zero since $(0,T)$ has finite measure. Then approximate that finite union by $\mathcal{A}$. Thus $\mathcal{M}$ is a sigma-algebra and contains all Borel sets. For a simple function on a finite measurable partition, approximate its cells by $\mathcal{A}$, disjointify the approximating cells, and assign the complement to the last cell. The new and old partitions disagree only inside the union of the cellwise symmetric differences. Replace each value by a nearby element of $D$. The integral squared error is bounded by a constant times the measure of that disagreement plus $T$ times the maximal squared value error. It tends to zero. The resulting functions take values in $D$ and have finitely many rational breakpoints within $(0,T)$; there are only countably many of them. They are dense, proving separability.