I encountered the following statement in the context of the Hecke algebra on the space of cusp forms, left without proof:
Let there be a family of commutative self-adjoint operators on a finite dimensional vector space $V$. Then there exists a basis of $V$ consisting of functions which are eigenfunctions for all the operators.
I would appreciate a proof of this, as elementary as possible, or a reference. I only know basic functional analysis and no spectral theory so I don't know where to start.