As a math student, I am also interested in the historical development of ideas in math.
I am studying Conway's Functional analysis book, and he uses the notation: $\sigma(X,X^*)$ for the weak topology on $X$ and $\sigma(X^*,X)$ for the weak-* topology on $X^*$. I think this sigma notation is pretty standard so it makes me wonder.
So my question: is the sigma notation $\sigma(X,X^*)$ or $\sigma(X^*,X)$ an artifact of an older math concept applied to functional analysis? Or is it simply just that, notation?