Let us consider a basic one-dimensional Boundary Value Problem of the following form: given function $f : [0,1] \to\mathbb R$, the unknown solution $u : [0,1] \to\mathbb R$ satisfies $$ -u''(x)=f(x),\quad x \in(0,1),\quad u(0) = 0,\quad u'(1)=0. $$ Then we can pose the equivalent weak formulation: Find $u\in V$ such that $$a(u,v)= (f,v),\quad\forall v\in V$$ where the form of $a$ is not repeated, since it is very classical (see page 3 of Introduction to Finite Elements Methods). We can then discretize the above by searching $u_h\approx u$ in a finite-dimensional subset $V_h$ of $V$, that is find $u_h\in V_h$ such that $$a(u_h,v)= (f,v),\quad\forall v\in V_h$$ Classically, we consider $$u_h(x)= \sum_{i=1}^n u_i\phi_i(x)$$ and everything is fine when $n$ is finite. The question is on taking the limit of the above for convergence analysis purposes. Taking the limit reads (note that the dimension of $V_h$ has to increase accordingly): $$a\Bigl(\lim_{n\to\infty}\sum_{i=1}^n u_i\phi_i,v\Bigr)= (f,v),\quad\forall v\in V_h\tag{form 1}$$ which commonly becomes $$\lim_{n\to\infty}\sum_{i=1}^n u_i a(\phi_i,v)= (f,v),\quad\forall v\in V_h\tag{form 2}$$
What are the arguments in weak formulations guaranteeing that going from (form 1) to (form 2) is safe? Which type of convergence is expected? Only weak convergence? Because we should not forget that the above derivations implies the following:
- term-wise differentiation:
$$u'_h(x)=\Bigl(\sum_{i=1}^n u_i\phi_i(x)\Bigr)'=\sum_{i=1}^n u_i\phi'_i(x)$$
- integral-sum swap
$$a(u_h,v)=\int_0^1u'_h(x)v'(x)\mathrm{d}x=\int_0^1 \sum_{i=1}^n u_i\phi'_i(x) v'(x)\mathrm{d}x=\sum_{i=1}^n u_i\int_0^1\phi'_i(x) v'(x)\mathrm{d}x$$
