# 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. 1.

If $x,y\in S$ and $x\neq y$, then $x$ is orthogonal (http://planetmath.org/OrthogonalVector) to $y$.

2. 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.

Title orthonormal set OrthonormalSet 2013-03-22 12:07:24 2013-03-22 12:07:24 yark (2760) yark (2760) 14 yark (2760) Definition msc 65F25 OrthogonalPolynomials OrthonormalBasis orthonormal