Let $X$ and $Y$ be two topological vector spaces. Suppose that there exists a bilinear map $\langle\cdot,\cdot\rangle:X\times Y\rightarrow\mathbb{R}$ satisfying
(i) If $\langle x,y\rangle=0$ for all $y\in Y$ then $x=0$;
(ii) If $\langle x,y\rangle=0$ for all $x\in X$ then $y=0$.
By definition, each $y\in Y$ defines a linear function $\langle\cdot,y\rangle:X\rightarrow\mathbb{R}$ on $X$. My question is that can any linear function on $X$ be generated in this way? That is, suppose that $f:X\rightarrow\mathbb{R}$ is a linear function. Does there exist $y\in Y$ such that $f(x)=\langle x,y\rangle$ for all $x\in X$?
On the wiki of "dual system",
https://en.wikipedia.org/wiki/Dual_system#Weak_representation_theorem
I find a "weak representation theorem" that seems to be relevant to my question. But I do not quite understand $\sigma(X,Y,b)$ in the statement of the theorem. If I already endow $X$ and $Y$ topologies separately, what is the role of $\sigma(X,Y,b)$? Besides, I thought every linear function is continuous, so the answer to my question is yes?