How is Baire category theorem used here? The following is a doubt that arouse from reading this paper by Bandyopadhyay, Jarosz and Rao. Let $E$ be a Banach space and $E^{*}$ be its dual space. Let $e_{0}$ be an element of norm one in $E$ such that its associated state space $S_{e_{0}}=\\{f\in E^{*}:\|f\|=1=f(e_{0})\\}$ spans $E^{*}$. Such an element is called a 'unitary' element of a Banach space. Let $S=\text{conv}(S_{e_{0}}\cup-S_{e_{0}}\cup iS_{e_{0}}\cup-iS_{e_{0}})$. Since Span $S_{e_{0}}=E^{*}$, we get $E^{*}=\cup_{n=1}^{\infty}nS$. Now $S$ is weak*-compact and hence norm closed. Here, the authors say that from the Baire category theorem, it follows that $\exists K>0$ such that $E^{*}_{1}\subseteq KS$, where $E^{*}_{1}$ is the closed unit ball of $E^{*}$. Why does this follow? The Baire category theorem should give us that at least one $nS$ has non-empty interior. How does the above conclusion follow? I would be grateful for a hint! Thanks in advance.

The set $S$ is convex and symmetric, i.e., $-x \in S$ for all $x\in S$. If $x$ is an interior point of $S$ and $B(x,\varepsilon)\subseteq S$ (where $B(x,\varepsilon)$ is the ball around $x$ with radius $\varepsilon$) you get $B(0,\varepsilon/2)\subseteq S$ since for $\|y\| <\varepsilon/2$ you have $y= \frac 12 (-x) +\frac 12 (x+2y) \in \frac 12 S + \frac 12 S \subseteq S$.