TITLE: Show that $B$ it is unbounded
QUESTION [5 upvotes]: Let $X$ a Banach space of infinity dimension. Show that any open set $B$, with $B \neq \emptyset$, is unbounded in the weak topology.

REPLY [9 votes]: Recall that if $X$ is a normed space, we can define continuous linear functionals (with respect to the given norm) and denote $X^*$ this collection. The weak topology is the most economic topology making continuous each elements of $X^*$.
If $O$ is an open subset containing $0$, then $0$ admits a basic neighborhood $V$contained in $O$, that is, an element of the form 
$$V=\bigcap_{j=1}^N\{x,|f_j(x)|<r\},$$
where $f_j\in X^*$ and $r$ is positive.  
Thus $V$ contains $\bigcap_{j=1}^N\ker f_j$, and so does $O$. 
Since $X$ is infinite dimensional we can find $x_0\neq 0$ such that $f_j(x_0)=0$ for all $j\in \{1,\dots,N\}$. Hence $O\supset \{\lambda x_0,\lambda\in\Bbb R\}$.