Measure, Integration & Real Analysis · Problem 10C.7

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hilbert-schmidt-norm-basis-independence

Problem

Suppose

T

is a bounded operator on a Hilbert space

V

. Prove that if

\{e_k\}_{k \in \Gamma}

and

\{f_j\}_{j \in \Omega}

are orthonormal bases of

V

, then

\sum_{k \in \Gamma} \|Te_k\|^2 = \sum_{j \in \Omega} \|Tf_j\|^2.