Every linear map on a finite dimensional complex vector space has an eigenvalue. Not so in the infinite case.
I'm interested in nice counterexamples anyone might have.
Consider the vector space $\mathbb C^\infty$ of sequences and the right shift map $R$ defined by
$$R(a_1, a_2, a_3, ...) = (0, a_1, a_2, a_3, ...)$$
$R$ has no eigenvalue (using the usual convention that there must be a non-trivial eigenvector).