An orthonormal basis of a finite-dimensional vector space 
with inner product 
is a vector basis 
whose vectors
are pairwise orthogonal and have norm
one. Thus 
for 
and 
.
An orthonormal set is linearly independent, so it is an orthonormal basis for the vector space it spans.
In a Hilbert space 
, an orthonormal basis means an orthonormal
set 
whose vector space span is dense
in 
.
This is also called a Hilbert basis. Equivalently,
every 
has the expansion
 |
where the sum is the norm limit of sums over finite subsets of 
. For an infinite indexing set, this differs from an algebraic
vector basis, which requires every vector
to be a finite linear combination of basis
vectors (Axler 2026, pp. 242-244).
The simplest example of an orthonormal basis is the standard basis 
for Euclidean space 
. The vector 
has all coordinates equal
to 0 except for a 1 in the 
th coordinate. For example,
.
In Euclidean space, a rotation
about the origin, or such a rotation
combined with a reflection through a hyperplane
containing the origin, sends an orthonormal
set to another orthonormal set. Conversely,
any orthonormal basis of 
can be sent to the standard
basis by a rotation, possibly combined with a reflection. These linear
transformations preserve the inner product and
are called orthogonal transformations.
It is often convenient to use an orthonormal basis for calculations. For example, the formula for a vector space projection is simpler in an orthonormal basis. Gram-Schmidt orthonormalization produces an orthonormal basis from a vector basis.
A real symmetric matrix has an orthonormal basis of eigenvectors. For a general matrix, the eigenvectors need not be orthonormal or even form a vector basis.
