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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
(*) ψ(''u'',''e<sub>j</sub>'')=∫<sub>T</sub>''
=== First discretization step ===
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