Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an automorphism, i.e. a bijective endomorphism. If $V$ is finite-dimensional, we know that the characteristic polynomial $\chi_f$ has a zero $\lambda$, which is an eigenvalue of $f$. However, does a similar argument hold if $\dim V=\infty$?
Remark: In my previous question, I asked the same without requiring $f$ to be bijective. The counterexamples from the answers weren't bijective, so now I'm wondering if there's a bijective counterexample as well or if there's a way to prove the statement for the bijective case.