I am reading a set of lecture notes on Functional Analysis and I have come across a not introduced notation when reading a corollary of Hahn-Banach Extension Theorem. Could anybody please explain what does $S_{X^*}$ represent?
The corollary is the following:
Let $X$ be a normed vector space. For each $x_0 \in X$ there exists $f \in S_{X^*}$ such that $f(x_0) = \| x_0\|$. In particular, $\|x\| = \max\{ |f(x)| : \; f \in S_{X^*}\}$ for all $x\in X$.
I do not have access to notes of prerequisite courses and it is difficult to get in touch with the writer of the document.
Any help appreciated!
For any normed space $X$, $S_X$ usually denotes the unit sphere of $X$: $$S_X=\{x\in X:\|x\|=1\}.$$ This result tells us that there is a functional $f\in X^*$ such that $|f(x)|\leq\|x\|$ for all $x\in X$, and $|f(x_0)|=\|x_0\|$.