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orthonormal basis (Definition)

Definition

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

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

\begin{displaymath} x=\sum_{b\in B}{\langle x,b\rangle}b. \end{displaymath}

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 ${\langle x,b\rangle}b$ 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
\begin{displaymath} {\langle x,y\rangle}=\sum_{b\in B}{\langle x,b\rangle}{\langle b,y\rangle}. \end{displaymath}



"orthonormal basis" is owned by yark. [ full author list (4) | owner history (5) ]
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See Also: Riesz sequence, orthonormal set, classification of Hilbert spaces

Other names:  Hilbert basis
Also defines:  dimension of a Hilbert space, dimension

Attachments:
every Hilbert space has an orthonormal basis (Theorem) by asteroid
all orthonormal bases have the same cardinality (Theorem) by asteroid
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Cross-references: unconditionally convergent, series, uncountable, expression, infinite, well-defined, Hilbert space, cardinality, every Hilbert space has an orthonormal basis, linear combinations, norm, dense in, linear span, orthonormal set, subset, inner product space
There are 52 references to this entry.

This is version 16 of orthonormal basis, born on 2003-10-15, modified 2008-03-21.
Object id is 5346, canonical name is OrthonormalBasis.
Accessed 10712 times total.

Classification:
AMS MSC: 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )
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