Measure, Integration & Real Analysis · Problem 8C.3

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · bessel-inequality-countable-support

Problem

Suppose

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

is an orthonormal family in an inner product space

V

. Prove that if

f \in V

, then

\{k \in \Gamma : \langle f, e_k\rangle \neq 0\}

is a countable set.