Definition:Orthonormal Subset

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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 if and only if:

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

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

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

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