Suppose $V$ is an inner-product space. Prove that if $T \in \mathcal{L}(V)$ is normal, then the minimal polynomial of $T$ has no repeated roots.
Does anyone know how to prove this for the case where $V$ is a real inner-product space? Proving it for a complex vector space is pretty easy using facts about normal operators but the same arguments don't apply for real inner-product spaces.