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Encyclopedia > Dual pair

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. ... The text or formatting below is generated by a template which has been proposed for deletion. ... 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 Eβ 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


  Results from FactBites:
 
Tolltex, Inc. Dual Tire Detection (871 words)
The dual tire sensors are shown in orange and are labled A1, B1, A2, and B2.
Distinguishing dual tires from single tires is based on the geometry of installing the dual tire sensors on angles to the curb and direction of vehicle travel.
Dual tires cause both sets of each dual tire sensor to be activated at the same time.
Dual topology - Wikipedia, the free encyclopedia (307 words)
In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.
Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.
The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of X', and the finest topology is the Mackey topology, the topology of uniform convergence on all weakly compact subsets of X'.
  More results at FactBites »

 
 

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