# Galerkin approximations for parabolic PDE weak solution, getting a uniform bound

(As usual $V \subset H$ are separable Hilbert spaces)

In a book I read this about existence of the solutions to parabolic PDEs:

the approximate solution $u_n(t)$ solves the equation $$(u_n', w_j) + (Au_n, w_j) = \langle f, w_j \rangle\tag{1}$$ for $j=1,...,N$. Here $w_j$ is the set of basis functions associated with the problem. We can write this as $$u_n' + Au_n = f$$ as an equality in $L^2(0,T;V').$

Surely this is not an equality in $L^2(0,T;V')$ but an equality in (I think) $L^2(0,T;V_n')$ where $V_n$ is the finite dimensional subset of $V$ spanned on $w_j$ for $j=1,...,N$? Because we only have that $(1)$ holds for everything in $V_n$ (by linearity).

The author then obtains a bound on $u_n'$ using this equality -- any elaboration would be useful.

I posted this in MSE (http://math.stackexchange.com/questions/418443/an-equality-in-l20-tv-weak-solution-to-pde-via-galerkin-approximations), please see for some discussion if you like.

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