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Almost Convergence and a Theorem -of G. M. Petersen

IP.com Disclosure Number: IPCOM000128250D
Original Publication Date: 1975-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 7 page(s) / 19K

Publishing Venue

Software Patent Institute

Related People

J. Swetits: AUTHOR [+3]

Abstract

A result of G. M. P-etersen states that a'sequence is almost convergent if and only if it is the sum of a bounded B limitable and a bounded C limitable sequence, whe~re B and C are specific summability matrices. It is shown that the necessity of the con-dition is false.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Almost Convergence and a Theorem -of G. M. Petersen,

by J. Swetits

Abstract

A result of G. M. P-etersen states that a'sequence is almost convergent if and only if it is the sum of a bounded B limitable and a bounded C limitable sequence, whe~re B and C are specific summability matrices. It is shown that the necessity of the con-dition is false.

The linear span of the set of all bounded sequences that are either B-limitable or C-limill-,able provides an example of a closed subspace of the space of bounded sequences which is not the inter-section of its containing separable _FK spaces.

1970 A. M. S. subject classificatJon:

Primary, 40D25; Secondary, 46A45.

Key words and p_h 9: ra s e, almost. convergence; slummabil--'.~t- ma'r-Tx; FK space; convargence Qom a _4 n

1 Introduction:

A sequence {x k I is said to be B limitable if

(Equation Omitted)

converges. fx ) is C limitable if

(Equation Omitted)

is convergent. The B and C limitations

(Equation Omitted)

methods are discussed by Goffman and Petersen 131.

In E53 Petersen states that a sequence is almost conver-gent if and only if it is the sum of a bounded B limitable and a bounded C limitable sequence. In this note we show that the necessity of the condition is false. We then adapt the technique used by Petersen in establishing sufficiency to show that if a separable FK space, E, contains all bounded B limitable and all bounded C limitable sequences, then E contains th-e space of almost convergent sequences.

2. Notation:

Old Dominion University Page 1 Dec 31, 1975

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Almost Convergence and a Theorem -of G. M. Petersen

w denotes the space of all complex sequences. aco denotes the space of sequences that are almost convergent

to 0 [4]. c 0 denotes the space of null sequences. Z denotes the space of absolutely convergent series.bv denotes the space of sequence of bounded variation, and bv 0 = bv n c 0 B i

denotes the space of bounded sequences that are B limitable to 0. C 0 denotes the space of bounded sequences that are C limitable to'O.

A vector subspace E, of w "Ls a sequence space. If E has a locally convex topology, T, then (E,
r) is a K-space provided that the linear functions

(Equation Omitted)

are continuous on E. If, in addition, (E, T) is complete and metrizable then (E. T) is called an FK- space. If X {x k and

(Equation Omitted)

then (E3 T) is an AK-space if P n x converges to x for each x c E.

If E_, F is a separated dual- pair of vector spaces, then cr(E. F) denotes the weak topology on E by F and T(E.1 F) denotes the Mackey topology on E by F.

3. Main result:

In order to show that- the necessity of Petersen's condition is false, we require

Proposition 3-1: If

(Equation Omitted)

then

(Equation Omitted)

Proof: Suppose x 6 B 0 Then

(Equation Omitted)

where

(Equation Omitted)

and

(Equation Omitted)

Old Dominion University Page 2 Dec 31, 1975

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Almost Convergence and a Theorem -of G. M. Petersen

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