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 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 basis is 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 as the usual concept of dimension for vector spaces.)

If $B$ is an orthonormal basis of a Hilbert space $H$ , then for every $x\in H$ we have$$ x=\sum_{b\in B}\ip{x,b}b.$$ 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 $\ip{x,b}b$ are non-zero (even if $B$ itself is uncountable), and if there are infinitely many non-zero terms the series is unconditionally convergent. For any $x,y\in H$ we also have$$ \ip{x,y}=\sum_{b\in B}\ip{x,b}\ip{b,y}.$$