# Definition:Orthonormal Subset

## Definition

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 if and only if:

$(1): \quad \forall u \in S: \norm u = 1$

where $\norm {\, \cdot \,}$ is the inner product norm.

$(2): \quad S$ is an orthogonal set:
$\forall u, v \in S: u \ne v \implies \innerprod u v = 0$