Finite-input Bochner L2 of a separable Hilbert space is Hilbert

For $U,W\in L_{\rho }^2(V)$, use the Borel representatives obtained by the null-set modification in [F1], and denote them again by $U,W$. Their norm functions and pairings are measurable. Pointwise Cauchy–Schwarz gives $\mathrm{\mid }(U(y),W(y){)}_V\mathrm{\mid }\le \mathrm{\parallel }U(y){\mathrm{\parallel }}_V\mathrm{\parallel }W(y){\mathrm{\parallel }}_V$. Integrating and applying [F3] shows that the product is integrable and the pairing is finite. Changing either representative on a null set does not change the integral. The pairing is bilinear and symmetric. If its value on $(U,U)$ is zero, then for each integer $n\ge 1$ the set $E_n=\{y:\mathrm{\parallel }U(y){\mathrm{\parallel }}_V\ge 1\mathrm{/}n\}$ has measure zero, since $0=\int \mathrm{\parallel }U{\mathrm{\parallel }}^2\ge \rho (E_n)\mathrm{/}{n}^2$; their countable union contains every point where $U\mathrm{\ne }0$. Thus the class of $U$ is zero, proving positive definiteness.

Let $(U_n)$ be Cauchy in the displayed norm. By [A1] select a subsequence $(U_{n_k})$ with $\mathrm{\parallel }{U}_{n_{k+1}}-U_{n_k}{\mathrm{\parallel }}_{L_{\rho }^2(V)}\le 2^{-k}$. By [A2], choose strongly measurable representatives for these countably many classes and simple-approximation witnesses for each representative; replace each representative by zero on its measurable null exceptional set, obtaining Borel versions of the same classes. Put $d_k(y)=\mathrm{\parallel }{U}_{n_{k+1}}(y)-U_{n_k}(y){\mathrm{\parallel }}_V$ and $s_m={\sum }_{k=1}^m{d}_k$. Minkowski [F4] gives $\mathrm{\parallel }{s}_m{\mathrm{\parallel }}_2\le {\sum }_{k=1}^m{2}^{-k}\le 1$. The functions $s_m^2$ increase, so monotone convergence [F4] yields $\int ({\lim }_m{s}_m{)}^{2}d\rho \le 1$. Therefore ${\sum }_k{d}_k(y)<\mathrm{\infty }$ outside a measurable null set; otherwise its square integral would be infinite on a set of positive measure.

Let $E=\{y:{\sum }_k{d}_k(y)<\mathrm{\infty }\}$, a Borel set of full measure by step 1.2. On $E$, $(U_{n_k}(y))$ is Cauchy in $V$ and hence has a limit by [F2]. Define $U(y)$ to be this limit on $E$ and zero on $E^c$. The Borel maps equal to $U_{n_k}$ on $E$ and zero on $E^c$ converge pointwise everywhere to $U$; their pointwise limit is Borel measurable (equivalently, distances to a countable dense subset of $V$ are Borel limits). By [F5], $U$ is strongly measurable. If $g_m={\sum }_{k\ge m}{d}_k$, then $g_m\downarrow 0$ almost everywhere, $g_m\le {\lim }_k{s}_k$, and the latter is in $L^2$ by step 1.2. Dominated convergence [F6] gives $\mathrm{\parallel }{g}_m{\mathrm{\parallel }}_2\to 0$, while $\mathrm{\parallel }U-U_{n_m}{\mathrm{\parallel }}_V\le g_m$ almost everywhere on $E$. Thus $U_{n_m}\to U$ in $L_{\rho }^2(V)$.

Since $(U_n)$ is Cauchy and its subsequence converges to $U$, the full sequence converges to $U$: for $\epsilon >0$, choose $N$ so $\mathrm{\parallel }{U}_n-U_m\mathrm{\parallel }<\epsilon \mathrm{/}2$ for $m,n\ge N$, then choose $n_k\ge N$ with $\mathrm{\parallel }{U}_{n_k}-U\mathrm{\parallel }<\epsilon \mathrm{/}2$; the triangle inequality gives $\mathrm{\parallel }{U}_n-U\mathrm{\parallel }<\epsilon$ for $n\ge N$. This proves completeness.

Let $U\in L_{\rho }^2(V)$ and choose a strongly measurable representative with simple approximants and measurable null exceptional set as in the definition. Change the representative and approximants to zero on that set; the resulting representative is measurable and the finite-valued simple approximants converge to it everywhere. Truncate each approximant to zero where its norm exceeds $2\mathrm{\parallel }U(y){\mathrm{\parallel }}_V$. The truncation set is measurable; the truncated functions remain finite-valued simple functions, converge to $U$ almost everywhere, and their squared errors are bounded by $9\mathrm{\parallel }U(y){\mathrm{\parallel }}_V^2$. Dominated convergence [F6] gives convergence in $L_{\rho }^2(V)$, proving density.