Weak topology

Remarks

$\sigma(X, Y)$ is the coarsest topology on $X$ for which every map $x \mapsto \langle x, y \rangle$, $y \in Y$, is continuous; it is a Hausdorff locally convex topology. A net $(x_i)$ converges to $x$ in $\sigma(X, Y)$ if and only if $\langle x_i, y \rangle \to \langle x, y \rangle$ for every $y \in Y$. The continuous linear forms on $(X, \sigma(X, Y))$ are exactly the maps $x \mapsto \langle x, y \rangle$ with $y \in Y$. Example: in $\ell^2$, the unit vectors $e_n$ converge to $0$ in $\sigma(\ell^2, \ell^2)$, since $\langle e_n, y \rangle = y_n \to 0$ for each $y \in \ell^2$. Non-example: on $\ell^2$, the norm topology makes every map $x \mapsto \langle x, y \rangle$ continuous. But it is not the coarsest such topology, so it is not the weak topology: the sequence $(e_n)$ converges weakly but not in norm.