Research Article A Characterization of Completeness via …downloads.hindawi.com/journals/aaa/2014/596384.pdf · Asymmetric functional analysis has become a research branch of analysis

Research ArticleA Characterization of Completeness via AbsolutelyConvergent Series and the Weierstrass Test in AsymmetricNormed Semilinear Spaces

N Shahzad1 and O Valero2

1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21859 Saudi Arabia2Departamento de Ciencias Matematicas e Informatica Universidad de las Islas Baleares Carretera de Valldemossa km 7507122 Palma de Mallorca Spain

Correspondence should be addressed to N Shahzad nshahzadkauedusa

Received 4 April 2014 Revised 9 June 2014 Accepted 13 June 2014 Published 10 July 2014

Academic Editor J J Font

Copyright copy 2014 N Shahzad and O Valero This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Asymmetric normed semilinear spaces are studied A description of biBanach left K-sequentially complete and Smyth completeasymmetric normed semilinear spaces is provided and three appropriate notions of absolute convergence in the asymmetric normedframework are introduced Some characterizations of completeness are also obtained via absolutely convergent series Moreoveras an application a Weierstrass test for the convergence of series is derived

1 Introduction

Asymmetric normed linear spaces were applied to solveextremal problems arising in a natural way in mathematicalprogramming first by Duffin and Karlovitz in 1968 [1] andthen by Krein and Nudelman in 1977 [2] Since then theinterest in such asymmetric structures has been growingAsymmetric functional analysis has become a researchbranch of analysis nowadays In 2013 Cobzas [3] publisheda monograph entitled ldquoFunctional Analysis in AsymmetricNormed Spacesrdquo which collects a large number of results inthe aforesaid research line On account of [3] a systematicand deep study of asymmetric normed linear spaces andother related structures such as asymmetric normed semi-linear spaces has been made by Romaguera and some of hiscollaborators Many of the aforesaid results can be found in[4ndash20]

Inspired by the intense research activity in the field underconsideration our purpose is to study some properties ofasymmetric normed semilinear spaces The main goal of thispaper is to delve into the relationship between completenessof asymmetric normed semilinear spaces and the absolute

convergence of series in such a way that the classical contextof Banach normed linear spaces can be recovered as a par-ticular case In particular we introduce three absolute con-vergence notions which are appropriate to describe biBanachasymmetric normed semilinear spaces left 119870-sequentiallycomplete asymmetric normed semilinear spaces and Smythcomplete asymmetric normed semilinear spaces Moreoveras an application of the developed theory we derive acriterion inspired by the celebrated Weierstrass 119872-test forthe convergence of series of asymmetric normed semilinearspace valued bounded mappings

2 Preliminaries

Throughout we will denote the set of real numbers the setof nonnegative real numbers and the set of positive integernumbers by R R+ and N respectively

According to [3] a semilinear space (or cone)119884 is a subsetof a linear space 119883 such that 119909 + 119910 isin 119884 and 120582 sdot 119909 isin 119884 forall 119909 119910 isin 119884 and 120582 isin R+ Clearly every linear space can beconsidered also as a semilinear space

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 596384 8 pageshttpdxdoiorg1011552014596384

2 Abstract and Applied Analysis

Following [3] an asymmetric norm on a linear space119883 isa function sdot 119883 rarr R+ satisfying the following conditionsfor all 119909 119910 isin 119883 and 120582 isin R+

(i) 119909 = minus 119909 = 0 hArr 119909 = 0

(ii) 120582 sdot 119909 = 120582119909

(iii) 119909 + 119910 le 119909 + 119910

A pair (119883 sdot ) is called an asymmetric normed linearspace provided that 119883 is a linear space and sdot is anasymmetric norm on 119883 Observe that if (119883 sdot ) is anasymmetric normed linear space then the function sdot 119904defined on 119883 by 119909119904(119909) = max119909 minus 119909 for all 119909 isin 119883is a norm on119883

On account of [11] an asymmetric normed semilinearspace (also called normed cone in [3]) is a pair (119884 sdot )where119884 is a semilinear space of an asymmetric normed linear space(119883 sdot ) The restriction of sdot to 119884 is also denoted by sdot

In our context by a quasimetric space we mean a pair(119883 119889) such that 119883 is a nonempty set and 119889 is a function 119889 119883 rarr R+ satisfying the following conditions for all 119909 119910 119911 isin119883

(i) 119889(119909 119910) = 119889(119910 119909) = 0 hArr 119909 = 119910

(ii) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911)

Furthermore every quasimetric 119889 allows defining a met-ric 119889119904 on 119883 by 119889119904(119909 119910) = max119889(119909 119910) 119889(119910 119909) for all 119909 119910 isin119883

It is well known that given a quasimetric space (119883 119889)a topology T(119889) can be induced on 119883 which has as a basethe family of open 119889-balls 119861

119889(119909 119903) 119909 isin 119883 119903 gt 0 where

119861119889(119909 119903) = 119910 isin 119883 119889(119909 119910) lt 119903 for all 119909 isin 119883 and 119903 gt 0 We

will say that a subset 119860 of119883 is 119889-closed whenever 119860 is closedwith respect to the topology T(119889) Moreover if (119909

119899)119899isinN is a

sequence which converges to a point 119909 isin 119883 with respect tothe topologyT(119889) then we will say that (119909

119899)119899isinN 119889-converges

to 119909 Furthermore a quasimetric space (119883 119889) is said to bebicomplete provided that the associated metric space (119883 119889119904)is complete For a fuller treatment of quasimetric spaces werefer the reader to [21]

Each asymmetric norm sdot on a linear space119883 induces aquasimetric 119889

sdoton119883which is defined by 119889

sdot(119909 119910) = 119910minus119909

for all 119909 119910 isin 119883 According to [11] (see also [3]) an asymmet-ric normed linear space (119883 sdot ) is biBanach whenever theinduced quasimetric space (119883 119889

sdot) is bicomplete In addition

if 119884 is a semilinear space of a linear space 119883 then (119884 sdot )is called a biBanach asymmetric normed semilinear spaceprovided that the quasimetric space (119884 119889

sdot) is bicomplete

where the restriction of 119889sdot

to 119884 is also denoted by 119889sdot

Notice that given an asymmetric normed semilinearspace (119883 sdot ) 119889119904

sdot(119909 119910) = 119889

sdot119904(119909 119910) for all 119909 119910 isin 119883

In what follows we will work with asymmetric normedsemilinear spaces (119884 sdot ) in such a way that 119884 is assumed tobe a 119889

sdot-closed semilinear space of an asymmetric normed

linear space (119883 sdot) (where again the restriction of sdot to119884 isdenoted by sdot) Of course in order to work with asymmetricnormed linear spaces we only need to take 119884 = 119883

3 The Absolute Convergence of Series inAsymmetric Normed Linear Spaces

31TheNotion of Absolute Convergence of Series Let us recallthat a series in a normed linear space (119883 sdot ) is a sequenceof the form (sum

119899

119896=1119909119896)119899isinN where (119909

119899)119899isinN is a sequence in 119883

However it is customary to denote the sequence (sum119899119896=1119909119896)119899isinN

by suminfin119896=1119909119896 Of course when (sum119899

119896=1119909119896)119899isinN is convergent we

will say that the seriessuminfin119896=1119909119896is convergent and its limit will

be also denoted by suminfin119896=1119909119896[22]

In the classical framework of normed linear spaces thenotion of absolutely convergent series is stated as follows

If (119909119899)119899isinN is a sequence in a normed linear space (119883 sdot

) then the series suminfin119896=1119909119896is absolutely convergent provided

that the series of nonnegative real numbers suminfin119896=1119909119896 is

convergentThe preceding notion among other things allows char-

acterizing completeness of normed linear spaces [23]

Theorem 1 Let (119883 sdot ) be a normed linear space Then thefollowing assertions are equivalent

(1) (119883 sdot ) is Banach

(2) Every absolutely convergent series is 119889sdot-convergent

It is clear that the notion of series can be extended to theframework of asymmetric normed semilinear spaces in thefollowing obvious way [3]

A series in an asymmetric normed semilinear space(119883 sdot ) is a sequence of the form (sum119899

119896=1119909119896)119899isinN where (119909

119899)119899isinN

is a sequence in 119883 Following the notation used for series innormed linear spaces a series (sum119899

119896=1119909119896)119899isinN in an asymmetric

normed semilinear space will be also denoted by suminfin119896=1119909119896 Of

course when (sum119899119896=1119909119896)119899isinN is 119889

sdot-convergent we will say that

suminfin

119896=1119909119896is 119889sdot-convergent and its limit will be also denoted by

suminfin

119896=1119909119896

A natural attempt to extend the notion of absolute con-vergence to the asymmetric normed semilinear spaces wouldbe as follows

Definition 2 Let (119909119899)119899isinN be a sequence in asymmetric nor-

med semilinear space (119883 sdot)Then wewill say that the seriessum119899

119896=1119909119896is absolutely convergent provided that the series

suminfin

119896=1119909119896 is convergent Moreover (119883 sdot ) will be said to

have the absolute convergence property provided that everyabsolutely convergent series is 119889

sdot-convergent

We must point out that the absolute convergence wasintroduced by Cobzas in [3] in the framework of asymmetricnormed linear spaces (in fact in the context of asymmetricseminormed linear spaces) So in Definition 2 we extendthe absolute convergence to the case of asymmetric normedsemilinear spaces

Clearly the notion of absolutely convergent series innormed linear spaces is retrieved as a particular caseof the preceding one whenever the asymmetric norm inDefinition 2 is exactly a norm

Abstract and Applied Analysis 3

The example below shows that there exists an asymmetricnormed semilinear space without the absolute convergenceproperty

Example 3 Consider the asymmetric normed linear space(1198970 sdot +

sup) where 1198970 denotes the space of all real sequenceswith only a finite number of nonzero terms sdot +sup is theasymmetric norm defined by

119909+

sup = sup119899isinN

10038171003817100381710038171199091198991003817100381710038171003817max (1)

for all 119909 isin 1198970 and sdot max R rarr R+ is the asymmetric norm

defined by 119909max = max119909 0 for all 119909 isin R Now take thesequence (119909

119899)119899isinN in 119897

0given by

119909119899= (0 0 0

119899

⏞⏞⏞⏞⏞⏞⏞1

2119899 0 0 ) (2)

for all 119899 isin N It is clear that the seriessuminfin119896=1119909119896+

sup convergesHowever the series suminfin

119896=1119909119896does not 119889

sdot+

sup-converge

In Example 5 we will show that there are asymmetricsemilinear spaces with the absolute convergence property

Now it seems natural to wonder whether the charac-terization provided by Theorem 1 remains valid when wereplace in its statement ldquoBanach normed linear spacerdquo withldquobiBanach asymmetric normed semilinear spacerdquo Thus thedesired result could be stated as follows

ldquoLet (119883 sdot ) be an asymmetric normed semilinear spaceThen the following assertions are equivalent

(1) (119883 sdot ) is biBanach(2) (119883 sdot ) has the absolute convergence propertyrdquo

Nevertheless Example 21 in Section 4 shows that such aresult is not true Accordingly we can conclude that the factthat the asymmetric normed semilinear space is biBanachdoes not guarantee that the absolute convergence propertyholds Since biBanach completeness implies that the metric119889sdot119904 is complete it seems obvious that the characterization of

biBanach completeness must require an additional propertymore restrictive than the absolute convergence propertywhich involves 119889

sdot119904 and thus in some sense the norm sdot 119904

Inspired by this fact we propose the following notion whichalso retrieves as a particular case the classical one

Definition 4 An asymmetric normed semilinear space (119883 sdot) will be said to have the strong absolute convergenceproperty provided that every absolutely convergent series is119889sdot119904-convergent

Clearly asymmetric normed semilinear spaces that holdthe strong absolute convergence property form a subclass ofthose satisfying the absolute convergence property

The next example shows that there are asymmetricnormed semilinear spaces which do not have the strongabsolute convergence property

Example 5 Consider the asymmetric normed linear space(R sdot max) where sdot max is the asymmetric norm intro-duced in Example 3 Then the metric 119889

sdot119904

maxis exactly the

Euclidean metric on R It follows that the quasimetric space(R 119889sdot119904

max) is bicomplete and thus the asymmetric normed

linear space (R sdot max) is biBanach Define the sequence(119909119899)119899isinN in R by 119909

119899= minus1 for all 119899 isin N It is clear that

suminfin

119896=1119909119896is absolutely convergent since 119909

119899max = 0 for all 119899 isin

N However the series suminfin119896=1119909119896is not 119889

sdot119904

max-convergent So

(R sdot max) does not have the strong absolute convergenceproperty

The following is an example of an asymmetric normedsemilinear space with the strong absolute convergence prop-erty

Example 6 Consider the asymmetric normed linear space(R sdot max) introduced in Example 5 It is clear that the pair(R+ sdot max) is an asymmetric normed semilinear spaceSince 119909119904max = 119909 for all 119909 isin R

+ we have that every absolutelyconvergent series is 119889

sdot119904

max-convergent

In the remainder of this sectionwe consider a fewnotionsof completeness that arise in a natural way in the asymmetriccontext Thus we focus our efforts on characterizing thoseasymmetric normed semilinear spaces that enjoy the (strong)absolute convergence property in terms of the aforemen-tioned notions of completeness

32 The Characterization In order to achieve our goal letus recall two notions of completeness for quasimetric spacesthe so-called left 119870-sequential completeness and the Smythcompleteness

A quasimetric space (119883 119889) is said to be left119870-sequentiallycomplete (Smyth complete) provided that every left 119870-Cauchy sequence is 119889-convergent (119889119904-convergent) where asequence (119909

119899)119899isinN is left 119870-Cauchy whenever for each 120598 gt 0

there exists 119896 isin N such that 119889(119909119899 119909119898) lt 120598 for all119898 ge 119899 ge 119896

According to [11 24] an asymmetric normed semilinearspace (119883 sdot ) is left 119870-sequentially complete (Smyth com-plete) if the quasimetric space (119883 119889

sdot) is left 119870-sequentially

complete (Smyth complete)

321 Absolute Convergence In this subsection we providea description of asymmetric normed semilinear spaces thathave the absolute convergence property

The following result will be crucial for our purpose whoseproof can be found in [25]

Lemma 7 Let (119883 119889) be a quasimetric space and let (119909119899)119899isinN be

a left119870-Cauchy sequence If there exists a subsequence (119909119899119896

)119896isinN

of (119909119899)119899isinN which 119889-converges to 119909 then (119909

119899)119899isinN 119889-converges to

119909

From the preceding result we immediately obtain thefollowing one for asymmetric normed semilinear spaces

Lemma 8 Let (119883 sdot ) be an asymmetric normed semilinearspace and let (119909

119899)119899isinN be a left119870-Cauchy sequence If there exists


a subsequence (119909119899119896

)119896isinN of (119909

119899)119899isinN which 119889

sdot-converges to 119909

then (119909119899)119899isinN 119889sdot-converges to 119909

Taking into account the preceding lemma we charac-terize asymmetric normed semilinear spaces that enjoy theabsolute convergence property in the result below It must bepointed out that the equivalence between assertions (2) and(3) inTheorem 9 is given in [3] for asymmetric normed linearspaces However here we prove the equivalence following atechnique that although related to the one used in [3] allowsus to provide a bit more information in the spirit of [23]about the spaces under study (assertion (1) in the statementof Theorem 9)

Theorem 9 Let (119883 sdot) be an asymmetric normed semilinearspace Then the following assertions are equivalent

(1) suminfin119896=1119909119896is 119889sdot-convergent for every sequence (119909

119899)119899isinN in

119883 with 119909119899 lt 2minus119899 for all 119899 isin N

(2) (119883 sdot ) is left 119870-sequentially complete

(3) (119883 sdot ) has the absolute convergence property

Proof (1) rArr (2) Assume that (119909119899)119899isinN is a left 119870-Cauchy

sequence in (119883 119889sdot) Our aim is to show that (119909

119899)119899isinN

has a 119889sdot-convergent subsequence (119909

119899119896

)119896isinN Indeed we can

consider an increasing sequence (119899119896)119896isinN in N such that

119889sdot(119909119899119896

119909119904) lt 2minus119896 (3)

for all 119904 ge 119899119896 It follows that

119889sdot(119909119899119896minus1

119909119899119896

) lt 2minus119896minus1 (4)

for all 119896 isin N with 119896 gt 1 Hence we have that

119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1

with 10038171003817100381710038171003817119887119899119896119896minus110038171003817100381710038171003817lt 2minus119896minus1 (5)

for all 119896 isin N with 119896 gt 1 where 119887119899119896119896minus1

= 119909119899119896

minus 119909119899119896minus1

Nextdefine a sequence (119910

119899)119899isinN in119883 as follows

1199101= 1199091198991

119910119896= 119887119899119896119896minus1

forall119896 isin N with 119896 gt 1 (6)

It is obvious that 119910119896 lt 2

minus119896 for all 119896 isin N Then we obtainthat suminfin

119898=1119910119898is 119889sdot-convergent Since

119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1

= 119909119899119896minus2

+ 119887119899119896minus1119896minus2

+ 119887119899119896119896minus1

= 1199091198991

+ 11988711989921

+ sdot sdot sdot + 119887119899119896119896minus1

= 1199101+ sdot sdot sdot + 119910

119896

(7)

for all 119896 isin N we conclude that (119909119899119896

)119896isinN is 119889

sdot-convergent By

Lemma 8 the sequence (119909119899)119899isinN is 119889

sdot-convergent It follows

that the asymmetric normed semilinear space (119883 sdot ) is left119870-sequentially complete(2) rArr (3) Consider a sequence (119909

119899)119899isinN such that the

induced series sum119899119896=1119909119896is absolutely convergent Then given

120598 gt 0 there exists 1198990isin N such that suminfin

119896=119899+1119909119896 lt 120598 for all

119899 ge 1198990 Thus taking119898 119899 ge 119899

0such that119898 ge 119899 ge 119899

0 we have

that

119889sdot(

119899

sum

119896=1

119909119896

119898

sum

119896=1

119909119896) =

1003817100381710038171003817100381710038171003817100381710038171003817

119898

sum

119896=119899+1

119909119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119898

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817

le

infin

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817 lt 120598

(8)

It follows that (sum119899119896=1119909119896)119899isinN is a left119870-Cauchy sequence Since

(119883 sdot) is left119870-sequentially complete we obtain thatsuminfin119896=1119909119896

is 119889sdot-convergent(3) rArr (1) Consider a sequence (119909

119899)119899isinN such that 119909

119899 lt

2minus119899 for all 119899 isin N Then we show that the series sum119899

119896=1119909119896is

absolutely convergent Indeedsum119899119896=1119909119896 le sum

119899

119896=112119896lt 1 for

all 119899 isin N It follows that the series suminfin119896=1119909119896 is convergent

Thus by hypothesis we obtain that the series suminfin119896=1119909119896is 119889sdot-

convergent This concludes the proof

Next we give a few examples of asymmetric normedsemilinear spaces having the absolute convergence property

Example 10 In [24] the following asymmetric normed semi-linear spaces were proved to be left 119870-sequentially completeand thus by Theorem 9 all have the absolute convergenceproperty

(1) (119897infin sdot +

sup) and (119897+

infin sdot +

sup) where 119897infin is the set ofreal number sequences (119909

119899)119899isinN such that sup

119899isinN119909119899 lt

infin 119897+infin= 119909 isin 119897

infin 119909119899ge 0 for all 119899 isin N and sdot +sup

is the asymmetric norm introduced in Example 3

(2) (119897119901 sdot +119901) and (119897+

119901 sdot +119901) where 1 le 119901 lt infin 119897

119901is

the set of real number sequences (119909119899)119899isinN such that

suminfin

119899=1|119909119899|119901lt infin 119897+

119901= 119909 isin 119897

119901 119909119899ge 0 for all 119899 isin N

and sdot +119901

is the asymmetric norm defined on 119897119901by

119909+119901= (

infin

sum

119899=1

10038171003817100381710038171199091198991003817100381710038171003817

119901

max)

1119901

(9)

for all 119909 isin 119897119901

(3) (119862([0 1]) sdot +infin) and (119862

+([0 1]) sdot

+infin) where

119862([0 1]) is the set of all functions 119891 [0 1] rarr R

which are continuous from ([0 1] | sdot |) into (R | sdot|) 119862+([0 1]) = 119891 isin 119862([0 1]) 119891(119909) ge

0 for all 119909 isin [0 1] and sdot +infin

is the asymmetricnorm on 119862([0 1]) given by

10038171003817100381710038171198911003817100381710038171003817+infin

= sup119909isin[01]

1003817100381710038171003817119891(119909)1003817100381710038171003817max (10)

for all 119891 isin 119862[0 1]

(4) (1198880 sdot +

sup) and (119888+

0 sdot +

sup) where 1198880 is the set of realnumber sequences (119909

119899)119899isinN which converge to 0 and

119888+

0= 119909 isin 119888

0 119909119899ge 0 for all 119899 isin N


(5) (R119898 sdot max119898) and (R119898

+ sdot max119898) where 119898 isin N

R119898 = (1199091 119909

119898) 1199091 119909

119898isin R R119898

+= 119909 isin

R119898 1199091ge 0 119909

119898ge 0 and sdot max119898 is the

asymmetric norm on R119898 defined by

119909max119898 = max1le119894le119898

10038171003817100381710038171199091198941003817100381710038171003817max (11)

for all 119909 isin R119898

322 Strong Absolute Convergence In this subsection weprovide a description of asymmetric normed semilinearspaces that have the strong absolute convergence property

In the following the well-known result below plays acentral role [3]



)119896isinN


119899)119899isinN 119889119904-converges

to 119909

As a direct consequence we obtain the next result


119899)119899isinN be a left 119870-Cauchy sequence in (119883 119889

sdot)

If there exists a subsequence (119909119899119896

)119896isinN of (119909

119899)119899isinN which 119889119904

sdot-

converges to 119909 then (119909119899)119899isinN 119889119904


Next we are able to provide a description of asymmetricnormed semilinear spaces having the strong absolute conver-gence property

Theorem13 Let (119883 sdot) be an asymmetric normed semilinearspace Then the following assertions are equivalent


119899)119899isinN in


(2) (119883 sdot ) is Smyth complete(3) (119883 sdot ) has the strong absolute convergence property

Proof (1) rArr (2) Assume (119909119899)119899isinN is a left 119870-Cauchy


119899)119899isinN has a

119889sdot119904-convergent subsequence (119909

119899119896

)119896isinN By the same method

as in the proof of (1) rArr (2) of Theorem 9 we can show theexistence of a sequence (119910

119899)119899isinN such that 119909

119899119896

= 1199101+ sdot sdot sdot + 119910

119896

and 119910119896 lt 2minus119896 for all 119896 isin N Then by hypothesis the series

suminfin

119896=1119910119896is 119889119904sdot-convergent and hence as a consequence the

sequence (119909119899119896

)119896isinN is 119889

119904

sdot-convergent By Lemma 12 we obtain

that the sequence (119909119899)119899isinN is 119889119904

sdot-convergent So (119883 sdot ) is

Smyth complete(2) rArr (3) Consider a sequence (119909


induced series suminfin119896=1119909119896is absolutely convergent Then as in

the proof of (2) rArr (3) of Theorem 9 we have that thesequence (sum119899

119896=1119909119896)119899isinN is left 119870-Cauchy Since (119883 sdot ) is

Smyth complete we deduce that suminfin119896=1119909119896is 119889sdot119904-convergent

(3) rArr (1) Consider a sequence (119909119899)119899isinN such that

119909119899 lt 2

minus119899 for all 119899 isin N Then according to the proof of(3) rArr (1) of Theorem 9 we have that the series sum119899

119896=1119909119896is

absolutely convergent Since (119883 sdot ) has the strong absolute

convergence property we conclude that the series suminfin119896=1119909119896is

119889sdot119904-convergent

Among all asymmetric normed semilinear spaces givenin Example 10 the only ones that are Smyth completeare (119897+119901 sdot +119901) (119888+0 sdot +

sup) and (R119898

+ sdot max119898) (see [26])

Hence by Theorem 13 the aforesaid asymmetric normedsemilinear spaces have the strong absolute convergenceproperty

Since every Cauchy sequence is a left119870-Cauchy sequenceand every Smyth complete asymmetric normed semilinearspace is at the same time left 119870-sequential complete andbiBanach we immediately deduce the next result

Corollary 14 Let (119883 sdot ) be an asymmetric normed semilin-ear space which has the strong absolute convergence propertyThen the following assertions hold

(1) (119883 sdot ) is left 119870-sequentially complete(2) (119883 sdot ) is biBanach

Remark 15 Observe that Example 10 yields instances ofasymmetric normed semilinear spaces that hold the absoluteconvergence property but not the strong one Thereforeleft 119870-sequential completeness is not equivalent to thestrong absolute convergence property Moreover althoughthe strong absolute convergence property has been intro-duced in Definition 4 with the aim of describing biBanachcompleteness next we show that both the aforesaid notionsare not equivalent Indeed let (R sdot max) be the biBanachasymmetric normed linear space given in Example 5 In theaforementioned example it was shown that for the sequence(119909119899)119899isinN defined by 119909

119899= minus1 for all 119899 isin N the series suminfin

119896=1119909119896

is absolutely convergent however it is not 119889sdot119904

max-convergent

So (R sdot max) does not have the strong absolute convergenceproperty

In the light of the preceding remark it seems clear thatwe will need a new subclass of absolutely convergent seriesfor providing a characterization of biBanach asymmetricnormed semilinear spaces To this end we introduce thefollowing notion

Definition 16 If (119909119899)119899isinN is a sequence in an asymmetric

normed semilinear space (119883 sdot ) then the series suminfin119896=1119909119896

is 119904-absolutely convergent provided that the seriessuminfin119896=1119909119896119904

is convergent Moreover we will say that the asymmetricnormed semilinear space (119883 sdot ) has the 119904-absolute conver-gence propertywhenever every 119904-absolutely convergent seriesis 119889sdot119904-convergent

Obviously every 119904-absolutely convergent series is abso-lutely convergent and besides all spaces with the strongabsolute convergence property have at the same time the 119904-absolute convergence property However the converse doesnot hold Indeed (R sdot max) is an example of asymmetricnormed semilinear space which enjoys the 119904-absolute con-vergence property but not the strong absolute convergenceproperty


We end the section yielding the characterization in thecase of biBanach asymmetric normed semilinear spacesthrough the next result whose proof runs as the proof ofTheorem 1



119899)119899isinN in

119883 with 119909119899119904

lt 2minus119899 for all 119899 isin N

(2) (119883 sdot ) is biBanach(3) (119883 sdot ) has the 119904-absolute convergence property

4 The Weierstrass Test in AsymmetricNormed Semilinear Spaces

The relevance of Theorem 1 is given by its wide number ofapplications Among others it allows to state a criterionknown as Weierstrass 119872-test for the uniform convergenceof series of bounded mappings [27] For the convenience ofthe reader and with the aim of making our exposition self-contained let us recall the notion of bounded mapping [28]

Let (119883 119889) be a metric space and let 119878 be a nonempty setThen a mapping119891 119878 rarr 119883 is said to be bounded if for some119909 isin 119883 there exists119872

119909isin R+0such that 119889(119891(119904) 119909) le 119872

119909for all

119904 isin 119878 where R+0stands for the set of positive real numbers

Next consider a normed linear space (119883 sdot ) Thenthe set 119873119887(119878119883) = 119891 119878 rarr 119883 119891 is boundedbecomes a normed linear space endowed with the norm sdot infin

defined by 119891infin= sup

119904isin119878119891(119904) for all 119891 isin 119873119887(119878 119883)

Observe that the boundness of 119891 guarantees that 119891infinlt infin

Furthermore the normed linear space (119873119887(119878 119883) sdot infin) is

Banach provided that the normed linear space (119883 sdot ) is soThe following theorem contains theWeierstrass119872-test in

the normed case

Theorem 18 Let (119883 sdot ) be a Banach normed linear spaceand let 119878 be a nonempty set If (119891

119899)119899isinN is a sequence in119873119887(119878119883)

and there exists a sequence (119872119899)119899isinN in R+

0such that the series

suminfin

119896=1119872119896is convergent and 119891

119899infinle 119872119899for all 119899 isin N then the

series suminfin119896=1119891119896is 119889sdotinfin

-convergent

Our main goal in this section is to prove a version ofTheorem 18 in the realm of asymmetric normed semilinearspaces To this end Theorems 9 and 13 will play a crucialrole Of course the notion of bounded mapping can beformulated in our context in two different ways Indeed givena quasimetric space (119883 119889) and a nonempty set 119878 a mapping119891 119878 rarr 119883will be said to be bounded from the left if for some119909 isin 119883 there exists119872

119909isin R+ such that 119889(119891(119904) 119909) le 119872

119909for all

119904 isin 119878 A mapping 119891 119878 rarr 119883 is bounded from the right if forsome 119909 isin 119883 there exists119872

119909isin R+ such that 119889(119909 119891(119904)) le 119872

119909

for all 119904 isin 119878It is routine to check that the pair (119860119873119887

+(119878 119883) sdot

infin)

is an asymmetric normed semilinear space whenever (119883 sdot) is an asymmetric normed semilinear space where119860119873119887+(119878 119883) = 119891 119878 rarr 119883 119891 is bounded from the left

and the asymmetric norm sdot infin

is defined as in the normed

caseThe same occurs with (119860119873119887minus(119878 119883) sdot

infin)which can be

defined duallyIt is clear that the notions of bounded mapping in the

quasimetric case allow us to recover the bounded notionfor metric spaces Moreover 119860119873119887

+(119878 119883) = 119860119873119887

minus(119878 119883) =

119873119887(119878 119883) when (119883 sdot ) is a normed linear space and inaddition we impose the constraint ldquo119872

119909isin R+0rdquo in the

definition of bounded mappingThe next example gives mappings which are bounded

from both the left and the right

Example 19 Let (R sdot max) be the asymmetric normedlinear space given in Example 5 Define the mapping 119891

0

R rarr R by

1198910(119909) =

119909 if 119909 gt 00 if 119909 le 0

(12)

Clearly

119889sdotmax

(1198910(119909) 0) =

1003817100381710038171003817minus1198910 (119909)1003817100381710038171003817max = 0 (13)

for all 119909 isin R Thus 1198910is a bounded mapping from the left

Next consider the asymmetric normed semilinear space(R+ sdot max) and define the mapping 119891

1 R+ rarr R by

1198911(119909) =

1

119909if 119909 gt 1

1 if 119909 le 1(14)

Obviously

119889sdotmax

(0 1198911(119909)) =

10038171003817100381710038171198911(119909)1003817100381710038171003817max le 1 (15)

for all 119909 isin R+ Hence we have that 1198911is a bounded mapping

from the right

Of course the sets 119860119873119887+(119878 119883) and 119860119873119887

minus(119878 119883) are

different in general as shown in the preceding example Infact 119891

0notin 119860119873119887

minus(119878 119883)

In the next result we discuss the left 119870-sequentialcompleteness and the Smyth completeness of119860119873119887

+(119878 119883) and

119860119873119887minus(119878 119883)

Theorem 20 Let (119883 sdot ) be an asymmetric normed semi-linear space and let 119878 be a nonempty set If (119883 sdot ) is Smythcomplete then (119860119873119887

+(119878 119883) sdot

infin) and (119860119873119887

minus(119878 119883) sdot

infin)

are both left119870-sequentially complete

Proof We only prove that (119860119873119887+(119878 119883) sdot

infin) is left 119870-

sequentially complete Similar arguments can be applied toprove that (119860119873119887

minus(119878 119883) sdot

infin) is left 119870-sequentially com-

pleteLet (119891

119899)119899isinN be a left 119870-Cauchy sequence in 119860119873119887

+(119878 119883)

Then given 120598 gt 0 there exists 1198990isin N such that

119889sdotinfin

(119891119899 119891119898) lt 120598 for all 119898 ge 119899 ge 119899

0 Hence

119889sdot(119891119899(119904) 119891119898(119904)) lt 120598 for all 119904 isin 119878 and for all 119898 ge 119899 ge 119899

0

It follows that for each 119904 isin 119878 (119891119899(119904))119899isinN is a left 119870-Cauchy

sequence in (119883 sdot ) Since (119883 sdot ) is Smyth complete wededuce that there exists 119909

119904isin 119883 such that (119891

119899(119904))119899isinN 119889sdot119904-

converges to 119909119904


Next define the mapping 119891 119878 rarr 119883 by 119891(119904) = 119909119904 Then

119891 isin 119860119873119887+(119878 119883) Indeed the fact that 119891

1198990

is bounded fromthe left yields the existence of 119909

1198911198990

isin 119883 and1198721199091198911198990

isin R+ suchthat 119889

sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990

for all s isin 119878 Thus we obtainthat

119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)

for all 119904 isin 119878 Moreover the existence of 1198991isin N is guaranteed

such that 1198991ge 1198990and in addition 119889

sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899

0 Thus we deduce that

119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)

for all 119899 ge 1198990and for all 119904 isin 119878 So we have shown that (119891

119899)119899isinN

is 119889sdotinfin

-convergent as claimed

The next example shows that there are left119870-sequentiallycomplete asymmetric normed semilinear spaces whoseasymmetric normed semilinear space of bounded mappingsis not left 119870-sequentially complete

Example 21 Consider the left 119870-sequentially asymmetricnormed linear space (119888

0 sdot +

sup) introduced in Example 10Of course it is routine to check that (119888

0 sdot +

sup) is not Smythcomplete Let 119878 = 1 Define the sequence (119891

119899)119899isinN given

by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+

sup = 11198992 for all 119899 isin N Hence 119891

119899belongs to

119860119873119887+(1 1198880) Moreover suminfin

119896=1119891119896infin

is convergent Neverthe-lesssuminfin

119896=1119891119896is not 119889

sdotinfin

-convergent Thus byTheorem 9 weconclude that (119860119873119887

+(1 1198880) sdot infin) is not left 119870-sequentially

complete Observe in addition that (119860119873119887+(1 1198880) sdot infin) is

biBanach and hence we have an example of a biBanachasymmetric normed linear space which does not have theabsolute convergence property as announced in Section 31

The following example provides a Smyth complete asym-metric normed semilinear space whose asymmetric normedsemilinear space of bounded mappings is not Smyth com-plete

Example 22 Consider the Smyth complete asymmetric nor-med linear space (119888+

0 sdot +

sup) given (see Example 5) and let119878 be as in Example 21 Take the sequence (119891

119899)119899isinN given in

Example 21 Then it is clear that 119891119899belongs to 119860119873119887

+(1 119888+

0)

for all 119899 isin N and that the series suminfin119896=1119891119896infin

is convergentHowever the series suminfin

119896=1119891119896is not 119889

||sdot||119904

infin

-convergent Thusby Theorem 13 we conclude that (119860119873119887

+(1R) sdot

infin) is not

Smyth complete

We end the paper proving the announcedWeierstrass testin the asymmetric framework

Theorem 23 Let (119883 sdot ) be a Smyth complete asymmetricnormed semilinear space and let 119878 be a nonempty set If(119891119899)119899isinN is a sequence in119860119873119887+(119878 119883) and there exists a sequence

(119872119899)119899isinN in R+ such that the series suminfin

119896=1119872119896is convergent and

119891119899infinle 119872119899for all 119899 isin N then the series suminfin

119896=1119891119896is 119889sdotinfin

-convergent

Proof Sincesum119899119896=1119891119899infinle sum119899

119896=1119872119899and the seriessuminfin

119896=1119872119896is

convergent we have that the seriessuminfin119896=1119891119896infinis convergent

The Smyth completeness of (119883 sdot ) provides the left 119870-sequential completeness of 119860119873119887

+(119878 119883) and by Theorem 9

we deduce that the series suminfin119896=1119891119896is 119889sdotinfin

-convergent

Using arguments similar to those in the proof of preced-ing result one can get the following result

Theorem 24 Let (119883 sdot ) be a Smyth complete asymmetricnormed semilinear space and let 119878 be a nonempty set If(119891119899)119899isinN is a sequence in119860119873119887minus(119878 119883) and there exists a sequence




119896=1119891119896is 119889sdotinfin

-convergent

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The first authoracknowledges with thanks DSR for financial support Thesecond author acknowledges the support from the SpanishMinistry of Economy and Competitiveness under grant noMTM2012-37894-C02-01 The authors thank the referees forvaluable comments and suggestions which improved thepresentation of this paper

References

[1] R J Duffin and L A Karlovitz ldquoFormulation of linear programsin analysis I Approximation theoryrdquo SIAM Journal on AppliedMathematics vol 16 pp 662ndash675 1968

[2] M G Krein and A A NudelmanTheMarkovMoment Problemand Extremal Problems American Mathematical Society 1977

[3] S Cobzas Functional Analysis in Asymmetric Normed SpacesBirkhauser Basel Switzerland 2013

[4] C Alegre ldquoContinuous operators on asymmetric normedspacesrdquo Acta Mathematica Hungarica vol 122 no 4 pp 357ndash372 2009

[5] C Alegre and I Ferrando ldquoQuotient subspaces of asymmetricnormed linear spacesrdquo Boletın de la Sociedad MatematicaMexicana vol 13 no 2 pp 357ndash365 2007

[6] C Alegre I Ferrando L M Garcıa-Raffi and E A SanchezPerez ldquoCompactness in asymmetric normed spacesrdquo Topologyand its Applications vol 155 no 6 pp 527ndash539 2008

[7] C Alegre J Ferrer and V Gregori ldquoQuasi-uniform structuresin linear latticesrdquo The Rocky Mountain Journal of Mathematicsvol 23 no 3 pp 877ndash884 1993


[8] C Alegre J Ferrer and V Gregori ldquoOn the Hahn-Banachtheorem in certain linear quasi-uniform structuresrdquoActaMath-ematica Hungarica vol 82 no 4 pp 325ndash330 1999

[9] J Ferrer V Gregori and C Alegre ldquoQuasi-uniform structuresin linear latticesrdquo The Rocky Mountain Journal of Mathematicsvol 23 no 3 pp 877ndash884 1993

[10] LMGarcıa-Raffi ldquoCompactness andfinite dimension in asym-metric normed linear spacesrdquo Topology and its Applications vol153 no 5-6 pp 844ndash853 2005

[11] L M Garcıa-Raffi and S Romaguera ldquoSequence spaces andasymmetric norms in the theory of computational complexityrdquoMathematical and Computer Modelling vol 36 no 1-2 pp 1ndash112002

[12] L M Garcıa-Raffi S Romaguera and E A Sanchez-PerezldquoThe bicompletion of an asymmetric normed linear spacerdquoActaMathematica Hungarica vol 97 no 3 pp 183ndash191 2002

[13] L M Garcıa-Raffi and R Sanchez-Perez ldquoThe dual space of anasymmetric normed linear spacerdquo Quaestiones MathematicaeJournal of the South African Mathematical Society vol 26 no 1pp 83ndash96 2003

[14] L M Garcıa-Raffi S Romaguera and E A Sanchez-PerezldquoThe supremum asymmetric norm on sequence spaces ageneral framework tomeasure complexity distancesrdquo ElectronicNotes in Theoretical Computer Science vol 74 pp 39ndash50 2003

[15] L M Garcıa Raffi S Romaguera and E A Sanchez PerezldquoWeak topologies on asymmetric normed linear spaces andnon-asymptotic criteria in the theory of complexity analysis ofalgorithmsrdquo Journal of Analysis and Applications vol 2 no 3pp 125ndash138 2004

[16] J Rodrıguez-Lopez and S Romaguera ldquoClosedness of boundedconvex sets of asymmetric normed linear spaces and theHausdorff quasi-metricrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 13 no 3 pp 551ndash562 2006

[17] S Romaguera E A Sanchez-Perez and O Valero ldquoA char-acterization of generalized monotone normed conesrdquo ActaMathematica Sinica vol 23 no 6 pp 1067ndash1074 2007

[18] S Romaguera E A Sanchez Perez and O Valero ldquoDominatedextensions of functionals and 119881-convex functions on cancella-tive conesrdquo Bulletin of the Australian Mathematical Society vol67 no 1 pp 87ndash94 2003

[19] S Romaguera and M Sanchis ldquoSemi-Lipschitz functions andbest approximation in quasi-metric spacesrdquo Journal of Approxi-mation Theory vol 103 no 2 pp 292ndash301 2000

[20] S Romaguera and M Schellekens ldquoDuality and quasi-normability for complexity spacesrdquo Applied General Topologyvol 3 no 1 pp 91ndash112 2002

[21] H P A Kunzi ldquoNonsymmetric distances and their associatedtopologies about the origins of basic ideas in the area ofasymmetric topologyrdquo in Handbook of the History of GeneralTopology C E Aull and R Lowen Eds vol 3 pp 853ndash968Kluwer Academic New York NY USA 2001

[22] E A Ok Real Analysis with Economic Applications PrincetonUniversity Press 2007

[23] G J O Jameson Topology and Normed Spaces Chapman andHall London UK 1974

[24] J J Conradie and M D Mabula ldquoConvergence and left-119870-sequential completeness in asymmetrically normed latticesrdquoActaMathematica Hungarica vol 139 no 1-2 pp 147ndash159 2013

[25] I L Reilly P V Subrahmanyam and M K VamanamurthyldquoCauchy sequences in quasi-pseudo-metric spacesrdquo Monat-shefte fur Mathematik vol 93 no 2 pp 127ndash140 1982

[26] M D Mabula Compactness in asymmetric normed lattices[PhD thesis] Department of Mathematics and Applied Math-ematics University of Cape Town 2012

[27] D H Griffel Applied Functional Analysis Dover 1985[28] J L Kelley I Namioka and W F Donoghue Jr Linear

Topological Spaces D Van Nostrand 1963

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Following [3] an asymmetric norm on a linear space119883 isa function sdot 119883 rarr R+ satisfying the following conditionsfor all 119909 119910 isin 119883 and 120582 isin R+

(i) 119909 = minus 119909 = 0 hArr 119909 = 0

(ii) 120582 sdot 119909 = 120582119909

(iii) 119909 + 119910 le 119909 + 119910

A pair (119883 sdot ) is called an asymmetric normed linearspace provided that 119883 is a linear space and sdot is anasymmetric norm on 119883 Observe that if (119883 sdot ) is anasymmetric normed linear space then the function sdot 119904defined on 119883 by 119909119904(119909) = max119909 minus 119909 for all 119909 isin 119883is a norm on119883

On account of [11] an asymmetric normed semilinearspace (also called normed cone in [3]) is a pair (119884 sdot )where119884 is a semilinear space of an asymmetric normed linear space(119883 sdot ) The restriction of sdot to 119884 is also denoted by sdot

In our context by a quasimetric space we mean a pair(119883 119889) such that 119883 is a nonempty set and 119889 is a function 119889 119883 rarr R+ satisfying the following conditions for all 119909 119910 119911 isin119883

(i) 119889(119909 119910) = 119889(119910 119909) = 0 hArr 119909 = 119910

(ii) 119889(119909 119911) le 119889(119909 119910) + 119889(119910 119911)

Furthermore every quasimetric 119889 allows defining a met-ric 119889119904 on 119883 by 119889119904(119909 119910) = max119889(119909 119910) 119889(119910 119909) for all 119909 119910 isin119883

It is well known that given a quasimetric space (119883 119889)a topology T(119889) can be induced on 119883 which has as a basethe family of open 119889-balls 119861

119889(119909 119903) 119909 isin 119883 119903 gt 0 where

119861119889(119909 119903) = 119910 isin 119883 119889(119909 119910) lt 119903 for all 119909 isin 119883 and 119903 gt 0 We

will say that a subset 119860 of119883 is 119889-closed whenever 119860 is closedwith respect to the topology T(119889) Moreover if (119909

119899)119899isinN is a

sequence which converges to a point 119909 isin 119883 with respect tothe topologyT(119889) then we will say that (119909

119899)119899isinN 119889-converges

to 119909 Furthermore a quasimetric space (119883 119889) is said to bebicomplete provided that the associated metric space (119883 119889119904)is complete For a fuller treatment of quasimetric spaces werefer the reader to [21]

Each asymmetric norm sdot on a linear space119883 induces aquasimetric 119889

sdoton119883which is defined by 119889

sdot(119909 119910) = 119910minus119909

for all 119909 119910 isin 119883 According to [11] (see also [3]) an asymmet-ric normed linear space (119883 sdot ) is biBanach whenever theinduced quasimetric space (119883 119889

sdot) is bicomplete In addition

if 119884 is a semilinear space of a linear space 119883 then (119884 sdot )is called a biBanach asymmetric normed semilinear spaceprovided that the quasimetric space (119884 119889

sdot) is bicomplete

where the restriction of 119889sdot

to 119884 is also denoted by 119889sdot

Notice that given an asymmetric normed semilinearspace (119883 sdot ) 119889119904

sdot(119909 119910) = 119889

sdot119904(119909 119910) for all 119909 119910 isin 119883

In what follows we will work with asymmetric normedsemilinear spaces (119884 sdot ) in such a way that 119884 is assumed tobe a 119889

sdot-closed semilinear space of an asymmetric normed

linear space (119883 sdot) (where again the restriction of sdot to119884 isdenoted by sdot) Of course in order to work with asymmetricnormed linear spaces we only need to take 119884 = 119883

3 The Absolute Convergence of Series inAsymmetric Normed Linear Spaces

31TheNotion of Absolute Convergence of Series Let us recallthat a series in a normed linear space (119883 sdot ) is a sequenceof the form (sum

119899

119896=1119909119896)119899isinN where (119909

119899)119899isinN is a sequence in 119883

However it is customary to denote the sequence (sum119899119896=1119909119896)119899isinN

by suminfin119896=1119909119896 Of course when (sum119899

119896=1119909119896)119899isinN is convergent we

will say that the seriessuminfin119896=1119909119896is convergent and its limit will

be also denoted by suminfin119896=1119909119896[22]

In the classical framework of normed linear spaces thenotion of absolutely convergent series is stated as follows

If (119909119899)119899isinN is a sequence in a normed linear space (119883 sdot

) then the series suminfin119896=1119909119896is absolutely convergent provided

that the series of nonnegative real numbers suminfin119896=1119909119896 is

convergentThe preceding notion among other things allows char-

acterizing completeness of normed linear spaces [23]

Theorem 1 Let (119883 sdot ) be a normed linear space Then thefollowing assertions are equivalent

(1) (119883 sdot ) is Banach

(2) Every absolutely convergent series is 119889sdot-convergent

It is clear that the notion of series can be extended to theframework of asymmetric normed semilinear spaces in thefollowing obvious way [3]

A series in an asymmetric normed semilinear space(119883 sdot ) is a sequence of the form (sum119899

119896=1119909119896)119899isinN where (119909

119899)119899isinN

is a sequence in 119883 Following the notation used for series innormed linear spaces a series (sum119899

119896=1119909119896)119899isinN in an asymmetric

normed semilinear space will be also denoted by suminfin119896=1119909119896 Of

course when (sum119899119896=1119909119896)119899isinN is 119889

sdot-convergent we will say that

suminfin

119896=1119909119896is 119889sdot-convergent and its limit will be also denoted by

suminfin

119896=1119909119896

A natural attempt to extend the notion of absolute con-vergence to the asymmetric normed semilinear spaces wouldbe as follows

Definition 2 Let (119909119899)119899isinN be a sequence in asymmetric nor-

med semilinear space (119883 sdot)Then wewill say that the seriessum119899

119896=1119909119896is absolutely convergent provided that the series

suminfin

119896=1119909119896 is convergent Moreover (119883 sdot ) will be said to

have the absolute convergence property provided that everyabsolutely convergent series is 119889

sdot-convergent

We must point out that the absolute convergence wasintroduced by Cobzas in [3] in the framework of asymmetricnormed linear spaces (in fact in the context of asymmetricseminormed linear spaces) So in Definition 2 we extendthe absolute convergence to the case of asymmetric normedsemilinear spaces

Clearly the notion of absolutely convergent series innormed linear spaces is retrieved as a particular caseof the preceding one whenever the asymmetric norm inDefinition 2 is exactly a norm





119909+


10038171003817100381710038171199091198991003817100381710038171003817max (1)



119899)119899isinN in 119897

0given by

119909119899= (0 0 0

119899

⏞⏞⏞⏞⏞⏞⏞1

2119899 0 0 ) (2)



119896=1119909119896does not 119889

sdot+

sup-converge













sdot119904

maxis exactly the





suminfin


119899max = 0 for all 119899 isin


sdot119904

max-convergent So





sdot119904

max-convergent













)119896isinN



119909





a subsequence (119909119899119896

)119896isinN of (119909

119899)119899isinN which 119889






119899)119899isinN in






119899)119899isinN


119899119896



119889sdot(119909119899119896

119909119904) lt 2minus119896 (3)


119889sdot(119909119899119896minus1

119909119899119896



119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1



= 119909119899119896

minus 119909119899119896minus1



1199101= 1199091198991

119910119896= 119887119899119896119896minus1





119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1

= 119909119899119896minus2

+ 119887119899119896minus1119896minus2

+ 119887119899119896119896minus1

= 1199091198991

+ 11988711989921


= 1199101+ sdot sdot sdot + 119910

119896

(7)


)119896isinN is 119889

sdot-convergent By







119896=119899+1119909119896 lt 120598 for all

119899 ge 1198990 Thus taking119898 119899 ge 119899

0such that119898 ge 119899 ge 119899

0 we have

that

119889sdot(

119899

sum

119896=1

119909119896

119898

sum

119896=1

119909119896) =

1003817100381710038171003817100381710038171003817100381710038171003817

119898

sum

119896=119899+1

119909119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119898

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817

le

infin

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817 lt 120598

(8)




119899)119899isinN such that 119909

119899 lt


119896=1119909119896is


119899

119896=112119896lt 1 for






(1) (119897infin sdot +

sup) and (119897+

infin sdot +



119899isinN119909119899 lt

infin 119897+infin= 119909 isin 119897



(2) (119897119901 sdot +119901) and (119897+


119901is


suminfin

119899=1|119909119899|119901lt infin 119897+

119901= 119909 isin 119897

119901 119909119899ge 0 for all 119899 isin N

and sdot +119901


119909+119901= (

infin

sum

119899=1

10038171003817100381710038171199091198991003817100381710038171003817

119901

max)

1119901

(9)

for all 119909 isin 119897119901

(3) (119862([0 1]) sdot +infin) and (119862

+([0 1]) sdot

+infin) where





10038171003817100381710038171198911003817100381710038171003817+infin

= sup119909isin[01]

1003817100381710038171003817119891(119909)1003817100381710038171003817max (10)

for all 119891 isin 119862[0 1]

(4) (1198880 sdot +

sup) and (119888+

0 sdot +



119888+

0= 119909 isin 119888

0 119909119899ge 0 for all 119899 isin N


(5) (R119898 sdot max119898) and (R119898


R119898 = (1199091 119909

119898) 1199091 119909

119898isin R R119898

+= 119909 isin

R119898 1199091ge 0 119909



119909max119898 = max1le119894le119898

10038171003817100381710038171199091198941003817100381710038171003817max (11)






)119896isinN


119899)119899isinN 119889119904-converges

to 119909




sdot)


)119896isinN of (119909

119899)119899isinN which 119889119904

sdot-






119899)119899isinN in





119899)119899isinN has a


119899119896



119899)119899isinN such that 119909

119899119896

= 1199101+ sdot sdot sdot + 119910

119896


suminfin


sequence (119909119899119896

)119896isinN is 119889

119904











119909119899 lt 2


119896=1119909119896is





sup) and (R119898

+ sdot max119898) (see [26])







119896=1119909119896


max-convergent












119899)119899isinN in

119883 with 119909119899119904







119909for all




119904isin119878119891(119904) for all 119891 isin 119873119887(119878 119883)




the normed case





suminfin




-convergent



119909for all



119909


+(119878 119883) sdot

infin)





infin)which can be



+(119878 119883) = 119860119873119887

minus(119878 119883) =






0

R rarr R by

1198910(119909) =

119909 if 119909 gt 00 if 119909 le 0

(12)

Clearly

119889sdotmax

(1198910(119909) 0) =

1003817100381710038171003817minus1198910 (119909)1003817100381710038171003817max = 0 (13)



1 R+ rarr R by

1198911(119909) =

1

119909if 119909 gt 1

1 if 119909 le 1(14)

Obviously

119889sdotmax

(0 1198911(119909)) =

10038171003817100381710038171198911(119909)1003817100381710038171003817max le 1 (15)


from the right


minus(119878 119883) are


0notin 119860119873119887

minus(119878 119883)


+(119878 119883) and

119860119873119887minus(119878 119883)


+(119878 119883) sdot

infin) and (119860119873119887

minus(119878 119883) sdot

infin)





minus(119878 119883) sdot


pleteLet (119891


+(119878 119883)


119889sdotinfin

(119891119899 119891119898) lt 120598 for all 119898 ge 119899 ge 119899

0 Hence


0



119904isin 119883 such that (119891

119899(119904))119899isinN 119889sdot119904-





1198990


1198911198990

isin 119883 and1198721199091198911198990


sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990


119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)



sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899


119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)


119899)119899isinN

is 119889sdotinfin




0 sdot +


0 sdot +



by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+


119899belongs to


119896=1119891119896infin


119896=1119891119896is not 119889

sdotinfin







0 sdot +




+(1 119888+

0)



119896=1119891119896is not 119889

||sdot||119904

infin


+(1R) sdot

infin) is not

Smyth complete






119896=1119891119896is 119889sdotinfin

-convergent



119896=1119872119896is



+(119878 119883) and by Theorem 9


-convergent






119896=1119891119896is 119889sdotinfin

-convergent



Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119909+


10038171003817100381710038171199091198991003817100381710038171003817max (1)



119899)119899isinN in 119897

0given by

119909119899= (0 0 0

119899

⏞⏞⏞⏞⏞⏞⏞1

2119899 0 0 ) (2)



119896=1119909119896does not 119889

sdot+

sup-converge













sdot119904

maxis exactly the





suminfin


119899max = 0 for all 119899 isin


sdot119904

max-convergent So





sdot119904

max-convergent













)119896isinN



119909





a subsequence (119909119899119896

)119896isinN of (119909

119899)119899isinN which 119889






119899)119899isinN in






119899)119899isinN


119899119896



119889sdot(119909119899119896

119909119904) lt 2minus119896 (3)


119889sdot(119909119899119896minus1

119909119899119896



119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1



= 119909119899119896

minus 119909119899119896minus1



1199101= 1199091198991

119910119896= 119887119899119896119896minus1





119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1

= 119909119899119896minus2

+ 119887119899119896minus1119896minus2

+ 119887119899119896119896minus1

= 1199091198991

+ 11988711989921


= 1199101+ sdot sdot sdot + 119910

119896

(7)


)119896isinN is 119889

sdot-convergent By







119896=119899+1119909119896 lt 120598 for all

119899 ge 1198990 Thus taking119898 119899 ge 119899

0such that119898 ge 119899 ge 119899

0 we have

that

119889sdot(

119899

sum

119896=1

119909119896

119898

sum

119896=1

119909119896) =

1003817100381710038171003817100381710038171003817100381710038171003817

119898

sum

119896=119899+1

119909119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119898

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817

le

infin

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817 lt 120598

(8)




119899)119899isinN such that 119909

119899 lt


119896=1119909119896is


119899

119896=112119896lt 1 for






(1) (119897infin sdot +

sup) and (119897+

infin sdot +



119899isinN119909119899 lt

infin 119897+infin= 119909 isin 119897



(2) (119897119901 sdot +119901) and (119897+


119901is


suminfin

119899=1|119909119899|119901lt infin 119897+

119901= 119909 isin 119897

119901 119909119899ge 0 for all 119899 isin N

and sdot +119901


119909+119901= (

infin

sum

119899=1

10038171003817100381710038171199091198991003817100381710038171003817

119901

max)

1119901

(9)

for all 119909 isin 119897119901

(3) (119862([0 1]) sdot +infin) and (119862

+([0 1]) sdot

+infin) where





10038171003817100381710038171198911003817100381710038171003817+infin

= sup119909isin[01]

1003817100381710038171003817119891(119909)1003817100381710038171003817max (10)

for all 119891 isin 119862[0 1]

(4) (1198880 sdot +

sup) and (119888+

0 sdot +



119888+

0= 119909 isin 119888

0 119909119899ge 0 for all 119899 isin N


(5) (R119898 sdot max119898) and (R119898


R119898 = (1199091 119909

119898) 1199091 119909

119898isin R R119898

+= 119909 isin

R119898 1199091ge 0 119909



119909max119898 = max1le119894le119898

10038171003817100381710038171199091198941003817100381710038171003817max (11)






)119896isinN


119899)119899isinN 119889119904-converges

to 119909




sdot)


)119896isinN of (119909

119899)119899isinN which 119889119904

sdot-






119899)119899isinN in





119899)119899isinN has a


119899119896



119899)119899isinN such that 119909

119899119896

= 1199101+ sdot sdot sdot + 119910

119896


suminfin


sequence (119909119899119896

)119896isinN is 119889

119904











119909119899 lt 2


119896=1119909119896is





sup) and (R119898

+ sdot max119898) (see [26])







119896=1119909119896


max-convergent












119899)119899isinN in

119883 with 119909119899119904







119909for all




119904isin119878119891(119904) for all 119891 isin 119873119887(119878 119883)




the normed case





suminfin




-convergent



119909for all



119909


+(119878 119883) sdot

infin)





infin)which can be



+(119878 119883) = 119860119873119887

minus(119878 119883) =






0

R rarr R by

1198910(119909) =

119909 if 119909 gt 00 if 119909 le 0

(12)

Clearly

119889sdotmax

(1198910(119909) 0) =

1003817100381710038171003817minus1198910 (119909)1003817100381710038171003817max = 0 (13)



1 R+ rarr R by

1198911(119909) =

1

119909if 119909 gt 1

1 if 119909 le 1(14)

Obviously

119889sdotmax

(0 1198911(119909)) =

10038171003817100381710038171198911(119909)1003817100381710038171003817max le 1 (15)


from the right


minus(119878 119883) are


0notin 119860119873119887

minus(119878 119883)


+(119878 119883) and

119860119873119887minus(119878 119883)


+(119878 119883) sdot

infin) and (119860119873119887

minus(119878 119883) sdot

infin)





minus(119878 119883) sdot


pleteLet (119891


+(119878 119883)


119889sdotinfin

(119891119899 119891119898) lt 120598 for all 119898 ge 119899 ge 119899

0 Hence


0



119904isin 119883 such that (119891

119899(119904))119899isinN 119889sdot119904-





1198990


1198911198990

isin 119883 and1198721199091198911198990


sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990


119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)



sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899


119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)


119899)119899isinN

is 119889sdotinfin




0 sdot +


0 sdot +



by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+


119899belongs to


119896=1119891119896infin


119896=1119891119896is not 119889

sdotinfin







0 sdot +




+(1 119888+

0)



119896=1119891119896is not 119889

||sdot||119904

infin


+(1R) sdot

infin) is not

Smyth complete






119896=1119891119896is 119889sdotinfin

-convergent



119896=1119872119896is



+(119878 119883) and by Theorem 9


-convergent






119896=1119891119896is 119889sdotinfin

-convergent



Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










a subsequence (119909119899119896

)119896isinN of (119909

119899)119899isinN which 119889






119899)119899isinN in






119899)119899isinN


119899119896



119889sdot(119909119899119896

119909119904) lt 2minus119896 (3)


119889sdot(119909119899119896minus1

119909119899119896



119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1



= 119909119899119896

minus 119909119899119896minus1



1199101= 1199091198991

119910119896= 119887119899119896119896minus1





119909119899119896

= 119909119899119896minus1

+ 119887119899119896119896minus1

= 119909119899119896minus2

+ 119887119899119896minus1119896minus2

+ 119887119899119896119896minus1

= 1199091198991

+ 11988711989921


= 1199101+ sdot sdot sdot + 119910

119896

(7)


)119896isinN is 119889

sdot-convergent By







119896=119899+1119909119896 lt 120598 for all

119899 ge 1198990 Thus taking119898 119899 ge 119899

0such that119898 ge 119899 ge 119899

0 we have

that

119889sdot(

119899

sum

119896=1

119909119896

119898

sum

119896=1

119909119896) =

1003817100381710038171003817100381710038171003817100381710038171003817

119898

sum

119896=119899+1

119909119896

1003817100381710038171003817100381710038171003817100381710038171003817

le

119898

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817

le

infin

sum

119896=119899+1

10038171003817100381710038171199091198961003817100381710038171003817 lt 120598

(8)




119899)119899isinN such that 119909

119899 lt


119896=1119909119896is


119899

119896=112119896lt 1 for






(1) (119897infin sdot +

sup) and (119897+

infin sdot +



119899isinN119909119899 lt

infin 119897+infin= 119909 isin 119897



(2) (119897119901 sdot +119901) and (119897+


119901is


suminfin

119899=1|119909119899|119901lt infin 119897+

119901= 119909 isin 119897

119901 119909119899ge 0 for all 119899 isin N

and sdot +119901


119909+119901= (

infin

sum

119899=1

10038171003817100381710038171199091198991003817100381710038171003817

119901

max)

1119901

(9)

for all 119909 isin 119897119901

(3) (119862([0 1]) sdot +infin) and (119862

+([0 1]) sdot

+infin) where





10038171003817100381710038171198911003817100381710038171003817+infin

= sup119909isin[01]

1003817100381710038171003817119891(119909)1003817100381710038171003817max (10)

for all 119891 isin 119862[0 1]

(4) (1198880 sdot +

sup) and (119888+

0 sdot +



119888+

0= 119909 isin 119888

0 119909119899ge 0 for all 119899 isin N


(5) (R119898 sdot max119898) and (R119898


R119898 = (1199091 119909

119898) 1199091 119909

119898isin R R119898

+= 119909 isin

R119898 1199091ge 0 119909



119909max119898 = max1le119894le119898

10038171003817100381710038171199091198941003817100381710038171003817max (11)






)119896isinN


119899)119899isinN 119889119904-converges

to 119909




sdot)


)119896isinN of (119909

119899)119899isinN which 119889119904

sdot-






119899)119899isinN in





119899)119899isinN has a


119899119896



119899)119899isinN such that 119909

119899119896

= 1199101+ sdot sdot sdot + 119910

119896


suminfin


sequence (119909119899119896

)119896isinN is 119889

119904











119909119899 lt 2


119896=1119909119896is





sup) and (R119898

+ sdot max119898) (see [26])







119896=1119909119896


max-convergent












119899)119899isinN in

119883 with 119909119899119904







119909for all




119904isin119878119891(119904) for all 119891 isin 119873119887(119878 119883)




the normed case





suminfin




-convergent



119909for all



119909


+(119878 119883) sdot

infin)





infin)which can be



+(119878 119883) = 119860119873119887

minus(119878 119883) =






0

R rarr R by

1198910(119909) =

119909 if 119909 gt 00 if 119909 le 0

(12)

Clearly

119889sdotmax

(1198910(119909) 0) =

1003817100381710038171003817minus1198910 (119909)1003817100381710038171003817max = 0 (13)



1 R+ rarr R by

1198911(119909) =

1

119909if 119909 gt 1

1 if 119909 le 1(14)

Obviously

119889sdotmax

(0 1198911(119909)) =

10038171003817100381710038171198911(119909)1003817100381710038171003817max le 1 (15)


from the right


minus(119878 119883) are


0notin 119860119873119887

minus(119878 119883)


+(119878 119883) and

119860119873119887minus(119878 119883)


+(119878 119883) sdot

infin) and (119860119873119887

minus(119878 119883) sdot

infin)





minus(119878 119883) sdot


pleteLet (119891


+(119878 119883)


119889sdotinfin

(119891119899 119891119898) lt 120598 for all 119898 ge 119899 ge 119899

0 Hence


0



119904isin 119883 such that (119891

119899(119904))119899isinN 119889sdot119904-





1198990


1198911198990

isin 119883 and1198721199091198911198990


sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990


119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)



sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899


119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)


119899)119899isinN

is 119889sdotinfin




0 sdot +


0 sdot +



by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+


119899belongs to


119896=1119891119896infin


119896=1119891119896is not 119889

sdotinfin







0 sdot +




+(1 119888+

0)



119896=1119891119896is not 119889

||sdot||119904

infin


+(1R) sdot

infin) is not

Smyth complete






119896=1119891119896is 119889sdotinfin

-convergent



119896=1119872119896is



+(119878 119883) and by Theorem 9


-convergent






119896=1119891119896is 119889sdotinfin

-convergent



Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(5) (R119898 sdot max119898) and (R119898


R119898 = (1199091 119909

119898) 1199091 119909

119898isin R R119898

+= 119909 isin

R119898 1199091ge 0 119909



119909max119898 = max1le119894le119898

10038171003817100381710038171199091198941003817100381710038171003817max (11)






)119896isinN


119899)119899isinN 119889119904-converges

to 119909




sdot)


)119896isinN of (119909

119899)119899isinN which 119889119904

sdot-






119899)119899isinN in





119899)119899isinN has a


119899119896



119899)119899isinN such that 119909

119899119896

= 1199101+ sdot sdot sdot + 119910

119896


suminfin


sequence (119909119899119896

)119896isinN is 119889

119904











119909119899 lt 2


119896=1119909119896is





sup) and (R119898

+ sdot max119898) (see [26])







119896=1119909119896


max-convergent












119899)119899isinN in

119883 with 119909119899119904







119909for all




119904isin119878119891(119904) for all 119891 isin 119873119887(119878 119883)




the normed case





suminfin




-convergent



119909for all



119909


+(119878 119883) sdot

infin)





infin)which can be



+(119878 119883) = 119860119873119887

minus(119878 119883) =






0

R rarr R by

1198910(119909) =

119909 if 119909 gt 00 if 119909 le 0

(12)

Clearly

119889sdotmax

(1198910(119909) 0) =

1003817100381710038171003817minus1198910 (119909)1003817100381710038171003817max = 0 (13)



1 R+ rarr R by

1198911(119909) =

1

119909if 119909 gt 1

1 if 119909 le 1(14)

Obviously

119889sdotmax

(0 1198911(119909)) =

10038171003817100381710038171198911(119909)1003817100381710038171003817max le 1 (15)


from the right


minus(119878 119883) are


0notin 119860119873119887

minus(119878 119883)


+(119878 119883) and

119860119873119887minus(119878 119883)


+(119878 119883) sdot

infin) and (119860119873119887

minus(119878 119883) sdot

infin)





minus(119878 119883) sdot


pleteLet (119891


+(119878 119883)


119889sdotinfin

(119891119899 119891119898) lt 120598 for all 119898 ge 119899 ge 119899

0 Hence


0



119904isin 119883 such that (119891

119899(119904))119899isinN 119889sdot119904-





1198990


1198911198990

isin 119883 and1198721199091198911198990


sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990


119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)



sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899


119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)


119899)119899isinN

is 119889sdotinfin




0 sdot +


0 sdot +



by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+


119899belongs to


119896=1119891119896infin


119896=1119891119896is not 119889

sdotinfin







0 sdot +




+(1 119888+

0)



119896=1119891119896is not 119889

||sdot||119904

infin


+(1R) sdot

infin) is not

Smyth complete






119896=1119891119896is 119889sdotinfin

-convergent



119896=1119872119896is



+(119878 119883) and by Theorem 9


-convergent






119896=1119891119896is 119889sdotinfin

-convergent



Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119899)119899isinN in

119883 with 119909119899119904







119909for all




119904isin119878119891(119904) for all 119891 isin 119873119887(119878 119883)




the normed case





suminfin




-convergent



119909for all



119909


+(119878 119883) sdot

infin)





infin)which can be



+(119878 119883) = 119860119873119887

minus(119878 119883) =






0

R rarr R by

1198910(119909) =

119909 if 119909 gt 00 if 119909 le 0

(12)

Clearly

119889sdotmax

(1198910(119909) 0) =

1003817100381710038171003817minus1198910 (119909)1003817100381710038171003817max = 0 (13)



1 R+ rarr R by

1198911(119909) =

1

119909if 119909 gt 1

1 if 119909 le 1(14)

Obviously

119889sdotmax

(0 1198911(119909)) =

10038171003817100381710038171198911(119909)1003817100381710038171003817max le 1 (15)


from the right


minus(119878 119883) are


0notin 119860119873119887

minus(119878 119883)


+(119878 119883) and

119860119873119887minus(119878 119883)


+(119878 119883) sdot

infin) and (119860119873119887

minus(119878 119883) sdot

infin)





minus(119878 119883) sdot


pleteLet (119891


+(119878 119883)


119889sdotinfin

(119891119899 119891119898) lt 120598 for all 119898 ge 119899 ge 119899

0 Hence


0



119904isin 119883 such that (119891

119899(119904))119899isinN 119889sdot119904-





1198990


1198911198990

isin 119883 and1198721199091198911198990


sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990


119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)



sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899


119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)


119899)119899isinN

is 119889sdotinfin




0 sdot +


0 sdot +



by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+


119899belongs to


119896=1119891119896infin


119896=1119891119896is not 119889

sdotinfin







0 sdot +




+(1 119888+

0)



119896=1119891119896is not 119889

||sdot||119904

infin


+(1R) sdot

infin) is not

Smyth complete






119896=1119891119896is 119889sdotinfin

-convergent



119896=1119872119896is



+(119878 119883) and by Theorem 9


-convergent






119896=1119891119896is 119889sdotinfin

-convergent



Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198990


1198911198990

isin 119883 and1198721199091198911198990


sdot(1198911198990

(119904) 1199091198911198990

) le 1198721199091198911198990


119889sdot(119891 (119904) 119909

1198911198990

) le 119889sdot119904 (119891 (119904) 119891

1198990

(119904)) + 119889sdot(1198911198990

(119904) 1199091198911198990

)

le 120598 +1198721199091198911198990

(16)



sdot(119891(119904) 119891

1198991

(119904)) lt 120598 and119889sdot119904(1198911198991

(119904) 119891119899(119904)) lt 120598 for all 119899 ge 119899


119889sdot(119891 (119904) 119891

119899(119904)) le 119889

sdot(119891 (119904) 119891

1198991

(119904))

+ 119889sdot119904 (1198911198991

(119904) 119891119899(119904)) lt 2120598

(17)


119899)119899isinN

is 119889sdotinfin




0 sdot +


0 sdot +



by 119891119899(1) = (1119899

2 11198992 1119899

2 0 0 ) Clearly 119891

119899infin=

119891119899(1)+


119899belongs to


119896=1119891119896infin


119896=1119891119896is not 119889

sdotinfin







0 sdot +




+(1 119888+

0)



119896=1119891119896is not 119889

||sdot||119904

infin


+(1R) sdot

infin) is not

Smyth complete






119896=1119891119896is 119889sdotinfin

-convergent



119896=1119872119896is



+(119878 119883) and by Theorem 9


-convergent






119896=1119891119896is 119889sdotinfin

-convergent



Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Characterization of Completeness via …downloads.hindawi.com/journals/aaa/2014/596384.pdf · Asymmetric functional analysis has become a research branch of analysis