orthonormal basisOrthonormalBasis2003-10-15 01:32:212008-03-21 14:09:02Definitiondimension of a Hilbert spacedimension\def\ip#1{{\langle #1\rangle}}
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\PMlinkescapeword{finite}
\PMlinkescapeword{terms}
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\section*{Definition}
An \emph{orthonormal basis} (or \emph{Hilbert basis})
of an inner product space $V$
is a subset $B$ of $V$ satisfying the following two properties:
\begin{itemize}
\item $B$ is an orthonormal set.
\item The linear span of $B$ is dense in $V$.
\end{itemize}
The first condition means that all elements of $B$ have norm $1$
and every element of $B$ is \PMlinkname{orthogonal}{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 $B$.
\section*{Orthonormal bases of Hilbert spaces}
Every Hilbert space has an orthonormal basis.
The cardinality of this orthonormal basis
is called the \emph{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
\PMlinkname{the usual concept of dimension for vector spaces}{Dimension2}.)
If $B$ is an orthonormal basis of a Hilbert space $H$,
then for every $x\in H$ we have
\[
x=\sum_{b\in B}\ip{x,b}b.
\]
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
\[
\ip{x,y}=\sum_{b\in B}\ip{x,b}\ip{b,y}.
\]