Английская Википедия:Beta-dual space

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

In functional analysis and related areas of mathematics, the beta-dual or Шаблон:Math-dual is a certain linear subspace of the algebraic dual of a sequence space.

Definition

Given a sequence space Шаблон:Mvar, the Шаблон:Math-dual of Шаблон:Mvar is defined as

<math>X^{\beta}:= \left \{ x \in\mathbb{K}^\mathbb{N}\ : \ \sum_{i=1}^{\infty} x_i y_i\text{ converges }\quad \forall y \in X \right \}.</math>

Here, <math>\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}</math> so that <math>\mathbb{K}</math> denotes either the real or complex scalar field.

If Шаблон:Mvar is an FK-space then each Шаблон:Mvar in Шаблон:Math defines a continuous linear form on Шаблон:Mvar

<math>f_y(x) := \sum_{i=1}^{\infty} x_i y_i \qquad x \in X.</math>

Examples

• <math>c_0^\beta = \ell^1</math>
• <math>(\ell^1)^\beta = \ell^\infty</math>
• <math>\omega^\beta = \{0\}</math>

Properties

The beta-dual of an FK-space Шаблон:Mvar is a linear subspace of the continuous dual of Шаблон:Mvar. If Шаблон:Mvar is an FK-AK space then the beta dual is linear isomorphic to the continuous dual.