\begin{definition}[Definition:Orthonormal Subset]
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $S \subseteq V$ be a subset of $V$.
Then $S$ is an '''orthonormal subset''' {{iff}}:
:$(1): \quad \forall u \in S: \norm u = 1$
where $\norm {\, \cdot \,}$ is the inner product norm.
:$(2): \quad S$ is an orthogonal set:
::$\forall u, v \in S: u \ne v \implies \innerprod u v = 0$
\end{definition}