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# orthonormal set

# Definition

An *orthonormal set* is a subset $S$ of an inner product space,
such that ${\left\langle x,y\right\rangle}=\delta_{{xy}}$ for all $x,y\in S$.
Here ${\left\langle\cdot,\cdot\right\rangle}$ is the inner product,
and $\delta$ is the Kronecker delta.

More verbosely, we may say that an orthonormal set is a subset $S$ of an inner product space such that the following two conditions hold:

1. If $x,y\in S$ and $x\neq y$, then $x$ is orthogonal to $y$.

2. If $x\in S$, then the norm of $x$ is $1$.

Stated this way, the origin of the term is clear: an orthonormal set of vectors is both orthogonal and normalized.

# Notes

Note that the empty set is orthonormal, as is a set consisting of a single vector of unit norm in an inner product space.

The columns (or rows) of a real orthogonal matrix form an orthonormal set. In fact, this is an example of an orthonormal basis.

# Applications

A standard application is finding an orthonormal basis for a vector space, such as by Gram-Schmidt orthonormalization. Orthonormal bases are computationally simple to work with.

## Mathematics Subject Classification

65F25*no label found*

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