I saw a theorem online and it is:
Let $X$ be a TVS and let $x\in X$. Then $\hat{x}\in(X^*,wk^*)^*$. Therefore, we have \begin{align*} \sigma(X^*,X)\subseteq\sigma(X^*, (X^*, wk^*)^*). \end{align*} Furthermore, if $X$ is a normed space, then \begin{align*} \sigma(X^*,(X^*, wk^*)^*)\subseteq\sigma(X^*, X^{**}), \end{align*} where $X^{**}$ is the second Banach dual of the normed space $X$.
And I know $\sigma(X^*, X^{**})$ stands for weak topology on $X^*$. But what $\sigma(X^*,(X^*, wk^*)^*)$ means here? Thank you!
For a TVS $Y$, and any subset $F\subset Y^*$ in its topological dual, the topology $\sigma(Y,F)$ is defined as the initial topology on $X$ with respect to the family $(f\colon Y\rightarrow \mathbb{C})_{f\in F}$.
Thus in order to make sense of $\sigma(X^*,(X^*,wk^*)^*)$, put $Y=X^*$ and try to view $(X^*,wk^*)^*$ as a subspace $F\subset Y^*$. Note that $X^*\hookrightarrow (X^*,wk^*)$ is continuous, thus dualizing gives a continuous injection $(X^*,wk^*)^*\hookrightarrow X^{**}=Y^*$. Define $F$ to be the image of this injection.