Stochastic Galerkin coercivity and Cea quasi-optimality

Each ${\psi }_{\alpha }$ is a continuous scalar polynomial. Approximate it pointwise by finite-valued measurable simple real functions; multiplying those functions by a fixed $v\in V$ gives simple $V$-valued maps converging to ${\psi }_{\alpha }v$. Thus ${\psi }_{\alpha }v$ is strongly measurable by [F5] and has Bochner norm $\mathrm{\parallel }{\psi }_{\alpha }{\mathrm{\parallel }}_{L^2(\rho )}\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }v{\mathrm{\parallel }}_V$ by [F4]. A finite basis of $V_h$ and finite $\mathrm{\Lambda }$ therefore generate ${\mathcal{V}}_{\mathrm{\Lambda },h}$ from finitely many members of $L_{\rho }^2(V)$, making it finite-dimensional and conforming.

The model form and load satisfy the boundedness and coercivity hypotheses in [F6] by [F1], and the ambient space is Hilbert by [F3]. The finite-dimensional subspace condition is step 1.1. Apply [F6] with $M=a^{\ast }$ and $\alpha =a_{\ast }$; it gives the unique discrete solution and $\mathrm{\parallel }u-U_{\mathrm{\Lambda },h}\mathrm{\parallel }\le (a^{\ast }\mathrm{/}{a}_{\ast }){\inf }_{W_h\in {\mathcal{V}}_{\mathrm{\Lambda },h}}\mathrm{\parallel }u-W_h\mathrm{\parallel }$. This is the claimed estimate.