I need to prove: If $S$ is a subset of a Hilbert space $H$, then the closed linear span of $S$ is all of $H$ iff $S^{\perp}=\{0\}$.
I'm confused how to characterize an element in $\overline{\text{span}[S]}$. Some links works for an orthonormal set, but the argument doesn't work in case of an arbitrary set $S$.
Best Answer
Use the theorem that if $V$ is a closed linear subspace of the Hilbert space $H$ then $H=V\oplus V^\perp$. Apply to $V=\overline{\text{Span}(S)}$. Then $V=H$ iff $V^\perp=0$, But $V^\perp=S^\perp$.