(1) The Weak Topology on $E$ is defined as the topology on $E$ with respect to the mappings $f:E \to \mathbb K$, $f \in E^*$. It is denoted $\sigma(E,E^*)$.
(2) The Weak* topology on $E^*$ is the topology on $E^*$ with respect to the mappings $x̂ : E^* \to \mathbb K$, where $x̂ (l) = l(x)$, $l \in E^*$. It is denoted by $\sigma(E^*,E)$.
Now I want to write them down explicitly:
(1) $\gamma_f := \{f^{-1}(V): V $ open in $\mathbb K\}$, $\gamma := \bigcup_{f \in E^*} \gamma_{f}$,
(2) $\chi_{x̂ } := \{x̂ ^{-1}(V): V $ open in $\mathbb K$ }
$\chi := \bigcup_{x̂ \in E^{**}} \chi_{x̂ }$.
Question: Are $\sigma(E^*,E)$ and $\sigma(E,E^*)$ the smallest topology containing $\gamma$ and $\chi$ respectively?