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From [[functional analysis]], we know that we do not in fact need to try every possible test function ''v'' in V. In fact, if E={''e''<sub>1</sub>,''e''<sub>2</sub>,...} is a subset of V so that its [[linear span]] is [[dense]] in V (in a suitable topology) then we can use only those functions as test functions. That is, we are now trying to find a solution to the problem
\textrm{(*) }\forall e_j \in \mathrm{E} : \psi\left(u,e_j\right)=\int_T g\cdot e_j.
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