orthonormal basis

Definition

An orthonormal basis (or Hilbert basis)of an inner product space $V$is a subset $B$ of $V$ satisfying the following two properties:

• $B$ is an orthonormal set.

• The linear span of $B$ is dense in $V$.

The first condition means that all elements of $B$ have norm $1$and every element of $B$ is orthogonal (http://planetmath.org/OrthogonalVectors) to every other element of $B$.The second condition says that every element of $V$ can be approximated arbitrarily closely by (finite) linear combinations of elements of $B$.

Orthonormal bases of Hilbert spaces

Every Hilbert space has an orthonormal basis.The cardinality of this orthonormal basisis called the dimension of the Hilbert space.(This is well-defined,as the cardinality does not depend on the choice of orthonormal basis.This dimension is not in general the same asthe usual concept of dimension for vector spaces (http://planetmath.org/Dimension2).)

If $B$ is an orthonormal basis of a Hilbert space $H$,then for every $x\in H$ we have

$$x=\sum _{b\in B}⟨x,b⟩b.$$

Thus $x$ is expressed as a (possibly infinite)“linear combination” of elements of $B$.The expression is well-defined,because only countably many of the terms $⟨x,b⟩b$ are non-zero(even if $B$ itself is uncountable),and if there are infinitely many non-zero termsthe series is unconditionally convergent.For any $x,y\in H$ we also have

$$⟨x,y⟩=\sum _{b\in B}⟨x,b⟩⟨b,y⟩.$$