every orthonormal basis is a parseval frame. but what about the converse in the finite dimensional case?
Let's say $H$ is a n-dimensional Hilbert space and $a_1,..,a_n$ a parseval frame. then, of course, $a_1,..,a_n$ is a basis, since every frame spans the whole space (or maybe the closure in the infinite dimensional case).
But is it also an orthonormal basis then? I mean it satisfies Parsevals identity by definition. Does anybody know how to prove or contradict that?
Greatings
Roman