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A pair of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164503.png" />, of elements of a (topological) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164504.png" /> and the dual (topological) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164505.png" />, respectively, which satisfies the conditions
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164506.png" /></td> </tr></table>
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A pair of sets $\{a_t\}$ and $\{\xi_t\}$, $t \in T$, of elements of a [[topological vector space]] $X$ and the dual (topological) space $X^*$, respectively, which satisfies the conditions
 +
$$
 +
\xi_t(a_s) = \langle \xi_t, a_s \rangle = 0
 +
$$
 +
if $t \ne s$ and $\ne 0$ if $t=s$ (here, $\langle {\cdot},{\cdot} \rangle$ is the canonical bilinear form coupling $X$ and $X^*$).  For instance, a biorthogonal system consists of a [[Schauder basis]] and the set formed by the expansion coefficients of $x$ in it. In a [[Hilbert space]] $H$ with scalar product $\langle {\cdot},{\cdot} \rangle$ and basis $\{a_t\}$ the set $\{b_s\}$ satisfying the condition
 +
$$
 +
\langle a_t, b_s \rangle = \delta_{st}
 +
$$
 +
where $\delta_{st} = 1$ if $s = t$ and $\delta_{st} = 0$ if $s \ne t$, is also a basis; it is said to be the basis dual to $\{a_t\}$ and, since $H = H^*$, the sets $\{a_t\}$ and $\{b_s\}$ form a biorthogonal system. In particular, a basis in $H$ is said to be an [[orthonormal system]] if it is dual to itself.
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164507.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164508.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b0164509.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645010.png" /> is the canonical bilinear form coupling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645012.png" />). For instance, a biorthogonal system consists of a Schauder basis and the set formed by the expansion coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645013.png" /> in it. In a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645014.png" /> with scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645015.png" /> and basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645016.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645017.png" /> satisfying the condition
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However, there also exist biorthogonal systems which do not even form a [[weak basis]]; an example is the set of functions $\exp(ikx)$, $k\in\mathbb{Z}$, $x\in\mathbb{R}$, in the space of continuous periodic functions with the norm $\Vert f \Vert = \sup |f(x)|$.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645018.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645022.png" />, is also a basis; it is said to be the basis dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645023.png" /> and, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645024.png" />, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645026.png" /> form a biorthogonal system. In particular, a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645027.png" /> is said to be orthonormal if its dual to itself.
 
 
 
However, there also exist biorthogonal systems which do not even form a weak basis; an example is the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645030.png" />, in the space of continuous periodic functions with the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016450/b01645031.png" />.
 

Latest revision as of 17:02, 12 January 2014


A pair of sets $\{a_t\}$ and $\{\xi_t\}$, $t \in T$, of elements of a topological vector space $X$ and the dual (topological) space $X^*$, respectively, which satisfies the conditions $$ \xi_t(a_s) = \langle \xi_t, a_s \rangle = 0 $$ if $t \ne s$ and $\ne 0$ if $t=s$ (here, $\langle {\cdot},{\cdot} \rangle$ is the canonical bilinear form coupling $X$ and $X^*$). For instance, a biorthogonal system consists of a Schauder basis and the set formed by the expansion coefficients of $x$ in it. In a Hilbert space $H$ with scalar product $\langle {\cdot},{\cdot} \rangle$ and basis $\{a_t\}$ the set $\{b_s\}$ satisfying the condition $$ \langle a_t, b_s \rangle = \delta_{st} $$ where $\delta_{st} = 1$ if $s = t$ and $\delta_{st} = 0$ if $s \ne t$, is also a basis; it is said to be the basis dual to $\{a_t\}$ and, since $H = H^*$, the sets $\{a_t\}$ and $\{b_s\}$ form a biorthogonal system. In particular, a basis in $H$ is said to be an orthonormal system if it is dual to itself.

However, there also exist biorthogonal systems which do not even form a weak basis; an example is the set of functions $\exp(ikx)$, $k\in\mathbb{Z}$, $x\in\mathbb{R}$, in the space of continuous periodic functions with the norm $\Vert f \Vert = \sup |f(x)|$.

How to Cite This Entry:
Biorthogonal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Biorthogonal_system&oldid=11290
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article