I don't understand some notation.
I know that if $\mathscr{X}$ is a LCS, then $(\mathscr{X}^*,\text{wk}^*)^* = \mathscr{X}$, where $\text{wk}^*$ denotes the weak-star topology on $\mathscr{X}^*$.
From the above fact, we can see that $\mathscr{X}$ is the dual of $(\mathscr{X}^*,\text{wk}^*)$, so $\mathscr{X}$ has a weak-star topology.
However, the book (Conway) denotes this weak-star topology as $\sigma((\mathscr{X}, \text{wk}^*),\mathscr{X}^*) = \sigma(\mathscr{X}, \mathscr{X}^*)$.
I don't understand why this book denotes like that.
As I understand it, the notation should be written as $$\sigma((\mathscr{X}^*,\text{wk}^*)^*,(\mathscr{X}^*,\text{wk}^*)) = \sigma(\mathscr{X},(\mathscr{X}^*,\text{wk}^*)).$$
What am I missing now?
Indeed, it looks like $\sigma((\mathscr{X}, \text{wk}^*),\mathscr{X}^*)$ resulted from too much "notational juggling" (as the author put it). What you wrote is correct. Although I would go further and say that for any two vector spaces $\mathscr X,\mathscr Y$ with a bilinear pairing $\langle x,y\rangle$ defined between them, $\sigma(\mathscr X, \mathscr Y)$ means the topology on $\mathscr X$ induced by the family of seminorms $p_y(x) = |\langle x,y\rangle|$, $y\in \mathscr Y$.
That is, we should recognize that the original topologies on $\mathscr X$ and $\mathscr Y$ do not matter; the topology $\sigma$ depends only on what what elements $\mathscr X$ and $\mathscr Y$ contain, and how they are paired by $\langle \cdot, \cdot \rangle$. So it's redundant to write $\sigma((\mathscr{X}, \text{wk}^*),\mathscr{X}^*)$, $\sigma((\mathscr{X}, \text{wk}),\mathscr{X}^*)$, or $\sigma((\mathscr{X}, \text{whatever}),\mathscr{X}^*)$ -- all of these are precisely $\sigma( \mathscr{X} ,\mathscr{X}^*)$.