With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider $$u' + Au = f$$ in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, then the Hilbert-Schmidt theory doesn't apply to $A$, i.e, its eigenfunctions do not form a orthonormal basis of $H$. So we just have a basis of $V$ and $H$ called $\{\varphi_j\}$.

How then can I obtain a bound on $\lVert u_m' \rVert_{L^2(0,T;V')}$? Here $u_m'$ is the finite-dimensional solution of $$\langle u_m', \varphi_j \rangle + \langle Au_m, \varphi_j \rangle = \langle f, \varphi_j \rangle$$ for $j=1,...,m.$

We can define a projection $P_m:H \to V_m$ with $$(P_m u - u, v_m) = 0 \qquad \text{for all $v_m \in V_m$}$$ which is clearly bounded from $H \to H$. But we require a bound on $P_m: V \to V$ since the bilinear form $A$ generates is bounded in $V \times V$.

Obtaining this bound is easy to do when the basis are the eigenfunctions. But I don't know how to obtain this bound in this circumstance when the basis is not the eigenfunctions.

Are there any alternate techniques to get this bound? Thank you.