orthonormal set
Definition
An orthonormal set^{} is a subset $S$ of an inner product space^{}, such that $\u27e8x,y\u27e9={\delta}_{xy}$ for all $x,y\in S$. Here $\u27e8\cdot ,\cdot \u27e9$ 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 $x\ne y$, then $x$ is orthogonal^{} (http://planetmath.org/OrthogonalVector) 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 GramSchmidt orthonormalization^{}. Orthonormal bases are computationally simple to work with.
Title  orthonormal set 

Canonical name  OrthonormalSet 
Date of creation  20130322 12:07:24 
Last modified on  20130322 12:07:24 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  14 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 65F25 
Related topic  OrthogonalPolynomials 
Related topic  OrthonormalBasis 
Defines  orthonormal 