An example such that the continuous function defined on the sequences space dose not have a fixed point.

Question

I want to find an example such that in this space:

$l^2 = [(x_n):x_n \in R, \sum^\infty_nx_n^2<\infty]$ with the $L^2$ norm,

a continuous function $f(x_n)$ maps the closed unit ball $B$ to itself does not have a fixed point.

I tried to define mapping $(x_n)\rightarrow (y_n)$ and show that it does not have a fixed point. I tried to define $y_1=(1-||x||)^.5$, but I stuck here.

Could please anyone give me some hint about how to give this example and how to prove it well-defined?

Answer 1

$f(x)=(\sqrt {1-\|x\|^{2}}, x_1,x_2,...)$ is such a function. I will leave the verification to you.