The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1]$, need not have a fixed point.
Know about the Brouwer fixed point Theorem on $\mathbb{R} ^n$ which states that if $ \bar{B}\subseteq \mathbb{R} ^n,$ closed and $f:\bar{B}\to \bar{B} $ is continuous, then there exists $x^*$ such that $f(x^*)=x^*.$
However, I sense that it's not true on $C[0,1]$ but I can't think of an example. Can anyone, please, provide an example? Thanks for your time!