Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator.
Now, $E$ is a infinity dimensional Banach space, and $\Omega=\{x\in E : ||x||=1\}$ .
If $A:\Omega\rightarrow \Omega$ is completely continuous and $L:\Omega\rightarrow \Omega, L(x)=-x$, then how to show that $A$ is not homotopy with $L$?
(Updated after modification of question)
If $A$ is completely continuous then the closure of its range does not include all of $\Omega,$ since the unit sphere is closed but not compact. This establishes that the degree of $A$ is $0.$