I am confused about the following two definitions.
Let $E$ and $F$ be vector spaces over the field $\mathbb{K}$.
- $F$ separates points of $E$ under the bilinear form $\langle\cdot,\cdot \rangle$ if, given $e\in E$ such that $\langle e,f \rangle=0$ for all $f\in F$, then $e=0$. (or equivalently, if for each $e\neq 0$ in $E$, there exists $f\in F$ such that $\langle e,f \rangle\neq 0$)
- $E$ is a $T_2$-space. (i.e., any two points have disjoint neighborhoods.)
According to the first line of the following link, it seems that 1. and 2. are equivalent. I have no idea how 1. implies 2. or 2. implies 1. (in 1., it seems there is no concept of neighborhoods)
http://mathworld.wolfram.com/WeakTopology.html
So, given this bilinear form which separates points on $E$, we define $\sigma(E,F)$ as the weakest topology on $E$ such that all maps $P_f: E \to K: P_f(e) = \langle e, f \rangle$ for all $f \in F$are continuous.
Then $E$ is Hausdorff: let $e_1 \neq e_2$ in $E$, then $e_1 - e_2 \neq 0$, so there is some $f \in F$ with $\langle (e_1-e_2),f\rangle \neq 0$, by the second formulation of the separating points condition.
But then $P_f(e_1 - e_2) = P_f(e_1) - P_f(e_2) \neq 0$, so $P_f(e_1) \neq P_f(e_2)$ in the field $\mathbb{K}$, which is Hausdorff (usually even metrisable) and so these points have disjoint open neighbourhoods $O_1$ resp. $O_2$ in $\mathbb{K}$. Then $e_i \in P_f^{-1}[O_i], i=1,2$ and these sets are in $\sigma(E,F)$ and disjoint as the $O_i$ are. So $E$ is $\sigma(E,F)$-Hausdorff.
The reverse implication is similar, try it out.