If we consider the closed unit ball $\overline{\mathcal{B}}$ of a normed vector space (infinite dimension) and a continuous map $\phi:\overline{\mathcal{B}}\to \overline{\mathcal{B}}$ does it have at least one fixed point ?
Notice that the closed unit ball in infinite dimension is not a compact set (according to Riesz's theorem) but could we apply the theorem even if the set is not compact ?
Is the compacity an important hypothesis ?
Thanks in advance !