I am going trough a funtional analysis course and to prove the geometric form of the Hahn-Banach theorem we need to prove that
$\sup_ { \ b \in B } f(b) < \sup_ { \ b \in B \\ \|x\| < \epsilon.} f(b+x)$
We are in a normed space X, B is a convex subset of X and $f : X\rightarrow \mathbb{R} $ is a linear funtion
Honestly im stuck in this step, I guess that I have to use the definition of norm of a functional (wich is $\|f\|= \left\{|f(x)| \ \ x \in X \mid \|x\| = 1 \right\} $)
Thanks in advance