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Difference between revisions of "Biorthogonal system"

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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
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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
 
\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
+
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}
 
\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 orthonormal if its dual to itself.
+
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 orthonormal 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 $\Vert f \Vert = \sup |f(x)|$.
+
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 $\Vert f \Vert = \sup |f(x)|$.

Revision as of 21:12, 11 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 orthonormal 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=31235
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article