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

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · orthonormal-family-finite-closed-subspace

Problem

Explain why 8.58 implies that if

\Gamma

is a finite set and

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

is an orthonormal family in a Hilbert space

V

, then

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

is a closed subspace of

V