Measure, Integration & Real Analysis · Problem 8C.12

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · converse-parseval-identity

Problem

Prove the converse of Parseval's identity. More specifically, prove that if

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

is an orthonormal family in a Hilbert space

V

and

\|f\|^2 = \sum_{k\in\Gamma} |\langle f, e_k\rangle|^2

for every

f \in V

, then

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

is an orthonormal basis of

V