In functional analysis and related areas of mathematics a **dual pair** or **dual system** is a pair of vector spaces with an associated bilinear form. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...
A common method in functional analysis, when studying normed vector spaces, is to analyze the relationship of the space to its continuous dual, the vector space of all possible continuous linear forms on the original space. A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed by a bilinear form. Using the bilinear form, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...
## Definition
A **dual pair** is a 3-tuple consisting of two vector space *X* and *Y* over the same (real or complex) field and a bilinear form The fundamental concept in linear algebra is that of a vector space or linear space. ...
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The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...
with and ## Example A vector space *V* together with its algebraic dual *V* ^{*} and the bilinear form defined as forms a dual pair. For each dual pair we can define a new dual pair with A sequence space *E* and its beta dual *E*^{β} with the bilinear form defined as In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ...
forms a dual pair.
## See also |