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Version 12 |
| \PMlinkescapeword{even} |
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| \PMlinkescapeword{finite} |
\PMlinkescapeword{finite} |
| \PMlinkescapeword{terms} |
\PMlinkescapeword{terms} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
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| \section*{Definition} |
\section*{Definition} |
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| An orthonormal basis of an inner product space $V$ |
An orthonormal basis of an inner product space $V$ |
| is a subset $B$ of $V$ satisfying the following two properties: |
is a subset $B$ of $V$ satisfying the following two properties: |
| \begin{itemize} |
\begin{itemize} |
| \item $B$ is an orthonormal set. |
\item $B$ is an orthonormal set. |
| \item The linear span of $B$ is dense in $V$. |
\item The linear span of $B$ is dense in $V$. |
| \end{itemize} |
\end{itemize} |
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| The first condition means that all elements of $B$ have norm $1$ |
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 $B$. |
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 $B$. |
The second condition says that every element of $V$ can be approximated arbitrarily closely by (finite) linear combinations of elements of $B$. |
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| \section*{Orthonormal bases of Hilbert spaces} |
\section*{Orthonormal bases of Hilbert spaces} |
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| Every Hilbert space has an orthonormal basis. |
Every Hilbert space has an orthonormal basis. |
| Moreover, if $B$ is an orthonormal basis of a Hilbert space $H$, |
Moreover, if $B$ is an orthonormal basis of a Hilbert space $H$, |
| then for every $x\in H$ we have |
then for every $x\in H$ we have |
| \[ |
\[ |
| x=\sum_{b\in B}\ip{x,b}b. |
x=\sum_{b\in B}\ip{x,b}b. |
| \] |
\] |
| Thus $x$ is expressed as a (possibly infinite) |
Thus $x$ is expressed as a (possibly infinite) |
| ``linear combination'' of elements of $B$. |
``linear combination'' of elements of $B$. |
| The expression is well-defined, |
The expression is well-defined, |
| because only countably many of the terms $\ip{x,b}b$ are non-zero |
because only countably many of the terms $\ip{x,b}b$ are non-zero |
| (even if $B$ itself is uncountable), |
(even if $B$ itself is uncountable), |
| and if there are infinitely many non-zero terms |
and if there are infinitely many non-zero terms |
| the series is unconditionally convergent. |
the series is unconditionally convergent. |
