Definition:Orthonormal Subset

Definition
Let $\left({V, \left\langle {\cdot, \cdot} \right\rangle}\right)$ be an inner product space.

Let $S \subseteq V$ be a subset of $V$.

Then $S$ is an orthonormal subset iff:


 * $(1): \quad \forall u \in S: \left\Vert{u}\right\Vert = 1$

where $\left\Vert{\cdot}\right\Vert$ is the inner product norm.


 * $(2): \quad S$ is an orthogonal set:
 * $\forall u, v \in S: u \ne v \implies \left\langle {u, v}\right\rangle = 0$

Also see

 * Definition:Basis (Hilbert Space)
 * Orthonormal Subset Extends to Basis