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A Characterization of a Class of Barrelled Sequence Spaces

IP.com Disclosure Number: IPCOM000128252D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 6 page(s) / 18K

Publishing Venue

Software Patent Institute

Related People

J. Swetits: AUTHOR [+3]

Abstract

Bennett and Kalton have characterized dense barrelled sub-spaces of an arbitrary FK space. It is shown that, if the space is assumed to be AK, then this characterization assumes a simpler and more explicit form.

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 44% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

A Characterization of a Class of Barrelled Sequence Spaces

by J. Swetits

Abstract

Bennett and Kalton have characterized dense barrelled sub-spaces of an arbitrary FK space. It is shown that, if the space is assumed to be AK, then this characterization assumes a simpler and more explicit form.

1. Introduction:

In a recent paper [41 Bennett and Kalton characterized dense,, barrelled subspaces of an arbitrary FK space, E. In this note, it is shown that if E is assumed to be an AK space, then the characterization assumes a simpler and more explicit form.

2. Definition and preliminaries:

w denotes the vector space of sequences of complex numbers. A subspace, E, of w is a K space if it is endowed with a locally convex topology, T, such that the linear functionals

(Equation Omitted)

are continuous. In addition, if T~ is complete and metrizable, then (E. T) is an FK space.

If

(Equation Omitted)

let

(Equation Omitted)

has the property that

(Equation Omitted)

for each

(Equation Omitted)

then (E, T) is called an AK space. f E is an FK-AK space then the dual of E may be identified

Old Dominion University Page 1 Dec 31, 1976

Page 2 of 6

A Characterization of a Class of Barrelled Sequence Spaces

with

converges

(Equation Omitted)

If F is a subspace of E ~ containing the space of sequences with only finitely many non-zero terms, ~, then E, F form a separated pair under the bilinear form

(Equation Omitted)

(Equation Omitted)

denote the weak, Mackey and strong topologies, respectively, on E by F (see, e.g., [71).

If

(Equation Omitted)

is an infinite matrix of complex numbers, the sequence

(Equation Omitted)

is defined by

(Equation Omitted)

(Equation Omitted)

where E is a given sequence space.

A class of barrelled spaces:

Theorem 3.1: Let

(Equation Omitted)

subspace of E containing

(Equation Omitted)

is barrelled in E if and mla if

(Equation Omitted)

(ii)

(Equation Omitted)

Old Dominion University Page 2 Dec 31, 1976

Page 3 of 6

A Characterization of a Class of Barrelled Sequence Spaces

(Equation Omitted)

(Equation Omitted)

(Equation Omitted)

by

(Equation Omitted)

(Equation Omitted)

If c denotes the space of convergent sequences, then c A includes E 0 . Since c A is an FK space [9, ch. 121, it follows from [4, Theorem 1] that c A includes E. Thus, for any x c E,

00 E t k x k converges. Consequently E includes E 0 Since the k=O reverse inclusion is satisfied, we have E 0 E

Let {a (n) I be a sequence in E that is

(Equation Omitted)

Cauchy.

If

(Equation Omitted)

is defined by a = a (n) . then c includes

(Equation Omitted)

Consequently, c A includes E, [4, Theorem 11. Condition (ii) now follows from the fact that E is G(E E) sequentially com-plete.

(Sufficiency): Let {a (n) I be a sequence in E that is

(Equation Omitted)

bounded. Let. m denote the space of bounded sequences, and define

(Equation Omitted)

is sequentially complete.

Proof: (Necessity) Let

and define

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