Let $H$ be a [Hilbert space](/page/Hilbert%20Space) and let $E \subset H$ be an orthonormal set. The following conditions are equivalent. 1. $E$ is complete. 2. $\overline{\operatorname{span}}(E)=H$. 3. For every $x \in H$, \begin{align*} \|x\|_H^2 = \sup_{F \subset E,\ F \text{ finite}} \sum_{e \in F}|(x,e)_H|^2. \end{align*} 4. For every $x \in H$, the net of finite partial sums \begin{align*} \sum_{e \in F}(x,e)_H e, \end{align*} indexed by finite subsets $F \subset E$, converges to $x$ in $H$.