I would like to proof the next claim:
Let $X$ a Banach space, $F\colon X\to X^*$ a linear continuous function, $$ \Gamma:=\{f\in (\mathcal{C}([0,1],X)\,:\, f(0)=f(1)=0\mbox{ and }\|f\|\leq 1\} $$ and fix $t\in [0,1]$, $y\in \Gamma$. Then $F(y(t))\in X^*$. How I can prove that $$ \inf\limits_{f\in\Gamma} \{ F(y(t))(f(t))\}= -\|F(y(t))\| $$ ? I tried it using Hahn-Banach, but maybe there is another easier way. Thanks in advance.
Edit: sorry, I forgot to put that F es linear... Thanks for the answers.