Affine diffusion forms are well-defined on Bochner L2

Choose finite-valued measurable simple maps $U_n,W_n$ converging respectively to $U,W$ almost everywhere, using [F2]. On every intersection of their finite partition sets, $y\mapsto a_y(U_n(y),W_n(y))$ is an affine continuous scalar function of $y$ by the displayed coefficient formula in [F1]. It is therefore measurable. On the common full-measure set where both approximations converge, continuity of the bounded bilinear form gives convergence to $a_y(U(y),W(y))$. Setting the scalar function to zero on the common measurable exceptional null set gives a measurable version; an arbitrary modification on that set need not be measurable. The integral uses this version, as in [F4].

For a finite-valued simple $W_n$ converging almost everywhere to $W$, the scalar function $F(y)(W_n(y))$ is measurable on each partition piece because $F(y)$ is affine continuous into the finite-dimensional span of $f_0,\dots ,f_d$. Its pointwise limit on the full-measure convergence set is $F(y)(W(y))$, so setting the limit to zero on the measurable exceptional null set gives a measurable version in the convention [F4]. Since $\mathrm{\mid }{y}_j\mathrm{\mid }\le R_j$, $\mathrm{\parallel }F(y){\mathrm{\parallel }}_{V^{\ast }}\le \mathrm{\parallel }{f}_0{\mathrm{\parallel }}_{V^{\ast }}+{\sum }_j{R}_j\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_{V^{\ast }}$; hence $F\in L_{\rho }^2(V^{\ast })$.

The pointwise bound [F1] gives $\mathrm{\mid }{a}_y(U(y),W(y))\mathrm{\mid }\le a^{\ast }\mathrm{\parallel }U(y){\mathrm{\parallel }}_V\mathrm{\parallel }W(y){\mathrm{\parallel }}_V$. Both norm functions are square-integrable, so [F3] makes this majorant integrable. The form integral is finite and depends only on the almost-everywhere classes by [F4].

Pointwise duality gives $\mathrm{\mid }F(y)(W(y))\mathrm{\mid }\le \mathrm{\parallel }F(y){\mathrm{\parallel }}_{V^{\ast }}\mathrm{\parallel }W(y){\mathrm{\parallel }}_V$. Apply [F3] to obtain the stated integrability and functional bound.

Integrating the pointwise coercivity bound [F1] gives $\mathcal{A}(U,U)\ge a_{\ast }\int \mathrm{\parallel }U(y){\mathrm{\parallel }}_V^{2}d\rho =a_{\ast }\mathrm{\parallel }U{\mathrm{\parallel }}_{\mathcal{H}}^2$. Integrating the pointwise continuity bound and applying [F3] as in step 2.1 gives $\mathrm{\mid }\mathcal{A}(U,W)\mathrm{\mid }\le a^{\ast }\mathrm{\parallel }U{\mathrm{\parallel }}_{\mathcal{H}}\mathrm{\parallel }W{\mathrm{\parallel }}_{\mathcal{H}}$. These estimates prove the remaining assertions.