I am trying to understand weak Topologies by reading John Conway's Course in Functional Analysis and he lists a bunch of theorems such as:
"If $X$ is LCS, $(X,wk)^{*} =X^{*}$"
which are getting very confusing and as he calls it, an exercise in notational juggling. To better understand this, I am trying to make my own example, but I am already getting lost.
Let $X$ be a normed space, then the dual, $X^{*}$ is a Banach space.
I've read that if $X$ is merely a locally convex space, then $X^{*}$ isn't necessarily a Banach space and so the only reasonable topology to endow $X^{*}$ with is the $wk^{*}$ topology. However, if $X$ is a normed space, it seems reasonable that we can endow $X^{*}$ with a norm topology.
If we consider $X$ normed, generate a topological space from it $X^{*}$ and endow $X^{*}$ with its norm topology, call it $Y:= (X^{*}, \lVert \cdot \rVert)$. Then suppose we take $Y$ as its own Banach space, momentarily forgetting about $X$ entirely. If we try to construct $Y^{*}$ and that use it to construct a $wk$ topology to endow $Y$ with, is $(Y,wk_{Y}) = (X^{*}, wk_{X}^{*})$?
Here $wk_{Y}$ denotes the weak topology on $Y$ which was constructed from the linear functionals on $Y$ and $wk_{X}^{*}$ is the weak star topology on $X$ as usual.
I don't necessarily require a proof, but I would like some insight as to what is going on. If I were to distill this into a single question, it would be whether or not I can just "add stars" in the naive way I want and have it all work out?
Edit: I think I created a weird notational blunder by defining $Y:= (X^{*},\lVert \cdot\rVert)$ then later trying to write $(Y,wk_{Y})$, but I don't know how else to explain what I am trying to do.