Completeness of an orthonormal system
= Completeness of an orthonormal system
{title2=$\overline{\operatorname{span}\{e_n\}}=H$}
An <orthonormal set> is complete in a <Hilbert space> when the <closure> of its <linear span> is the whole space. Equivalently, the only vector orthogonal to every member is zero: project any vector onto the closed span, and its remaining orthogonal component must vanish. This is completeness of an expansion system, distinct from completeness of the ambient metric space.