A representation theorem for norms in Hilbert space (Q1012957)

It is shown that a convex, balanced \(o\)-symmetric set in a complex Hilbert space \(H\) is the intersection of all ellipsoids centred at \(o\) containing the set. (An ellipsoid centred at \(o\) is defined to be a set of the form \(\{x\in H: \langle Ax,x\rangle\leq 1\}\) where \(A\) is some positive bounded operator on \(H\).) Equivalently, any norm \(|\cdot|\) equivalent to the norm \(\|\cdot\|\) of \(H\) has the representation \(|x|=\sup\langle Ax,x\rangle^{1/2}\) where the supremum is taken over all positive bounded operators \(A\) such that \(\langle Ax,x\rangle\leq|x|^2\) for all \(x\in H\). As expected, the proof uses the Hahn-Banach theorem.