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Biorthogonal system

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A pair of sets and , , of elements of a (topological) vector space and the dual (topological) space , respectively, which satisfies the conditions

if , and if (here, is the canonical bilinear form coupling and ). For instance, a biorthogonal system consists of a Schauder basis and the set formed by the expansion coefficients of in it. In a Hilbert space with scalar product and basis the set satisfying the condition

where if and if , is also a basis; it is said to be the basis dual to and, since , the sets and form a biorthogonal system. In particular, a basis in 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 , , , in the space of continuous periodic functions with the norm .

How to Cite This Entry:
Biorthogonal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Biorthogonal_system&oldid=31234
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article