In the finite element method, at a certain point we arrive at the following Galerkian problem where it is desired to find the solution $u_h \space \in V_h$ that solves the following equation:
$$ a(u_h,v_h)=L(v_h) \space \space \space \forall v_h \in V_h $$
where $a$ and $L$ are, respectively, a bilinear and linear operators. I cannot understand why is normally stated that it is enough to test against a set of basis functions $\Phi_i \in V_h$ (which are linearly combined to form $u_h$)and not against all functions $v\in V_h$
Thank you very much in advance and I hope you may help me understanding this issue.