In Evans' PDE Book, in the first Theorem of regularity of parabolic PDE's there consider the case where the coefficients $a^{ij},b^i,c$ of the uniformly parabolic operator (divergent form) $L$ coefficients are all smooth and don't depend on the time parameter $t$. Let $\{w_n\}$ be a basis of orthonormal eigenvectors of the operator $\Delta$. Finally, let $\{u_m\}$ be the sequence of Galerkin approximations for the equation $$ \begin{cases} \mathbf{u}_t + L \mathbf{u}= \mathbf{f} &\text{ in } U\times [0,T] \\ \mathbf{u} = 0 &\text{ in } \partial U \times [0,T]\\ \mathbf{u}(0) = g &\text{ in } U \end{cases} $$ where $\mathbf{f} \in H^{1}(0,T; L^2(U)) $, $g \in H^1_0(U)\cap H^2(U) $ and $U$ is a open, bounded subset of $\mathbb{R}^N $ with smooth boudary.
In p. 363 (of the first edition), Evans points out $$ (I) \qquad \|\mathbf{u}^\prime_{m}(0) \| \le C (\|\mathbf{f} \|_{H^{1}(0,T; L^2(U))} + \|\mathbf{u}(0) \|_{H^2(U)}) $$ (he actually claims something the term in the LHR plus something else is smaller than the RHS, so the term $\|\mathbf{f} \|_{H^{1}(0,T; L^2(U))}$ might not be required) which he justifies referring to the equation $$ (II)\qquad (\mathbf{u}^\prime_{m},w_k) + B[\mathbf{u}_m,w_k ]= (\mathbf{f},w_k). $$
Well, my attempt is to multiply $(II)$ by the right coefficients in both sides and evaluate at $0$ (which at this point I don't know how to justify that $(II)$ holds, as the equation only holds almost surely) to get $$ \|\mathbf{u}^\prime_{m}(0)\|^2_{L^2(U)} + B[\mathbf{u}_m,\mathbf{u}^\prime_m ]= (\mathbf{f},\mathbf{u}^\prime_m). $$ So I imagine the first step is to use the bounds of the coefficients of $L$ to get $$ (III) \qquad |B[\mathbf{u}_m(0),\mathbf{u}^\prime_m (0)]|\le C(\|\mathbf{u}_m(0)\|^2_{H^1_0(U)}+ \|\mathbf{u}^\prime_m(0) \|^2_{H^1_0(U)}) $$ and $$ (IV) \qquad |(\mathbf{f}(0),\mathbf{u}^\prime_m(0))|\le C(\|\mathbf{f}(0)\|^2_{L^2(U)}+ \|\mathbf{u}^\prime_m(0) \|^2_{L^2(U)}) $$
However, the fact that $(III)$ the RHS depends on $ \|\mathbf{u}^\prime_m(0) \|^2_{H^1_0(U)}$ and not on $ \|\mathbf{u}^\prime_m(0) \|^2_{L^2(U)}$ is rather annoying, as if it was the case, I could just use $\varepsilon$-Cauchy to get the desired result. Unfortunately, the Poincaré Inequality would point to the 'wrong' side of what I want. Moreover, as I said, I don't know how to justify that $(II)$ holds at $t=0$. How could I solve those problems?