I am interested in constructing the following "counter-example" to the Banach's fixed point theorem.

Let $K=$ {$ g\in L_1: \|g\|=1, g(\cdot)\ge0 $}. Clearly, $K$ is not a compact and $K$ is not closed.

My A question is: is it possible to construct a [edit] nonexpansive mapping $f: K\to K$ with no fixed point? ( i.e. a mapping $f$ such that [edit] for all $x\neq y\in K$ one has $\|f(x)-f(y)\| < \|x-y\|$. )