Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces.

I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that $$\langle u'(t), v(t) \rangle + b(u(t),v(t)) = \langle f(t), v(t) \rangle$$ for appropriate bilinear form $b$.

Is there a way to obtain an uniform bound on the derivative of the Galerkin approximations, $u_n'$, in the space $L^2(0,T;V^*)$ **without** using a orthogonal and orthonormal basis $w_j$ for the spaces $V$ and $H$ respectively? So I am looking for techniques to get
$$\lVert u'_n \rVert_{L^2(0,T;V^*)} < C$$
when the basis is not orthogonal/orthonormal.

(I know I can orthonormalise any basis but I don't want to do that).