Definition:Orthonormal Subset

Definition
Let $\left({V, \left\langle {\cdot} \right\rangle}\right)$ be an inner product space, and let $S \subseteq H$ be a subset of $H$.

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


 * $\forall u \in S: \left\Vert{u}\right\Vert = 1$, where $\left\Vert{\cdot}\right\Vert$ is the inner product norm.


 * $\forall u, v \in S: u \ne v \implies \left\langle {u, v} \right\rangle = 0$, that is: $S$ is an orthogonal set.