Definition:Orthonormal Subset

Definition
Let $H$ be a Hilbert space, and let $S \subseteq H$ be a subset of $H$.

Then $S$ is said to be an orthonormal subset iff:


 * $\forall e \in S: \left\|{e}\right\| = 1$
 * $\forall e, f \in S: e \ne f \implies e \perp f$

Basis
A basis for $H$ is a maximal orthonormal subset $S$ with respect to the subset ordering on the set of orthonormal subsets of $H$.