It is shown that a normalized Riesz basis for a Hilbert space H (i.e., the isomorphic image of an orthonormal basis in H) induces in a natural way a new, but equivalent, inner product on
H in which it is an orthonormal basis, thereby extending the
sense in which Riesz bases and orthonormal bases are thought of
as being the same. A consequence of the
method of proof of this result yields a series representation for
all positive isomorphisms on a Hilbert space.