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 !
Consider the following normed vector space:
$$V=\mathbb{R}\oplus \mathbb{R}\oplus\cdots$$ $$|v|=|(a_1,a_2,\ldots)|=\max(|a_1|, |a_2|, \ldots)$$
Note that $V$ is a direct sum, i.e. $V$ consists of sequences which are $0$ everywhere except for the finite number of indexes. Thus this is well defined and $V$ is a normed space.
Now let's define
$$\Phi:\mathcal{B}\to\mathcal{B}$$ $$\Phi(a_1, a_2, \ldots)=(1, a_1, a_2, \ldots)$$
The function is well defined (i.e. the image is in $\mathcal{B}$ because $|1|=1$) and continous but it does not have a fixed point. Indeed if $\Phi(v)=v$ for some $v=(a_1,a_2,\ldots)$ then by definition
$$a_1 = 1$$ $$a_2 = a_1 = 1$$ $$a_3 = a_2 = 1$$ $$\cdots$$ $$a_i = 1$$
In particular $v=(1, 1, \ldots)$ is a constant infinite sequence. It's a contradiction since sequences in $V$ have to be "finite" (in the sense having zeros almost everywhere).