The Swallow's Tail: Difference between revisions
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en>Fly by Night V = x^5 + ax^3 + bx^2 + cx is NOT the equation of the swallow tail. The swallow tail is the set of points (a,b,c) such that V has a repeated root. |
en>Nihiltres m Modernized infobox syntax |
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In [[functional analysis]] and related areas of [[mathematics]], the '''beta-dual''' or '''<math>\beta</math>-dual''' is a certain linear subspace of the [[algebraic dual]] of a [[sequence space]]. | |||
== Definition == | |||
Given a sequence space <math>X</math> the '''<math>\beta</math>-dual''' of <math>X</math> is defined as | |||
:<math>X^{\beta}:=\{x \in X \mid \sum_{i=1}^{\infty} x_i y_i < \infty \quad \forall y \in X\}.</math> | |||
If <math>X</math> is an [[FK-space]] then each <math>y</math> in <math>X^{\beta}</math> defines a [[continuous linear form]] on <math>X</math> | |||
:<math>f_y(x) := \sum_{i=1}^{\infty} x_i y_i \qquad x \in X.</math> | |||
== Examples == | |||
* <math>c_0^\beta = l^1</math> | |||
* <math>(l^1)^\beta = l^\infty</math> | |||
* <math>\omega^\beta = \emptyset</math> | |||
== Properties == | |||
The beta-dual of an FK-space ''E'' is a [[linear subspace]] of the [[continuous dual]] of ''E''. If ''E'' is an [[FK-AK space]] then the beta dual is linear isomorphic to the continuous dual. | |||
{{mathanalysis-stub}} | |||
[[Category:Functional analysis]] | |||
Revision as of 06:08, 16 January 2014
In functional analysis and related areas of mathematics, the beta-dual or -dual is a certain linear subspace of the algebraic dual of a sequence space.
Definition
Given a sequence space the -dual of is defined as
If is an FK-space then each in defines a continuous linear form on
Examples
Properties
The beta-dual of an FK-space E is a linear subspace of the continuous dual of E. If E is an FK-AK space then the beta dual is linear isomorphic to the continuous dual.