We have definition of weak topology and also of weak solution
but the thing which I do not understand is how they are related, for instance,
Let $X$ be a Banach space, then $x_n\rightharpoonup x$ iff $f(x_n)\to f(x), \forall~ f\in X^*$
But we say $u$ is a weak solution of $Lu=f$, where $L$ is any partial differential operator if $\forall \phi \in C_c^\infty(\Omega), ~u$ satisfies $$\langle Lu,\phi\rangle=\langle f, \phi\rangle$$ Does it says that $x=y$ in weak sense iff $f(x)=f(y)\forall~ f\in X^*$ ?