Research Article Spaces of Ideal Convergent …downloads.hindawi.com/journals/tswj/2014/134534.pdfSpaces of Ideal Convergent Sequences M.Mursaleen 1 andSunilK.Sharma 2 Department of
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Research ArticleSpaces of Ideal Convergent Sequences
M Mursaleen1 and Sunil K Sharma2
1 Department of Mathematics Aligarh Muslim University Aligarh 202002 India2Department of Mathematics Model Institute of Engineering amp Technology Kot Bhalwal JampK 181122 India
Correspondence should be addressed to M Mursaleen mursaleenmgmailcom
Received 20 August 2013 Accepted 26 November 2013 Published 28 January 2014
Academic Editors F Basar and J Xu
Copyright copy 2014 M Mursaleen and S K Sharma This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
In the present paper we introduce some sequence spaces using ideal convergence and Musielak-Orlicz function M = (119872119896) Wealso examine some topological properties of the resulting sequence spaces
1 Introduction and Preliminaries
The notion of ideal convergence was first introduced byKostyrko et al [1] as a generalization of statistical convergencewhich was further studied in topological spaces by Das et alsee [2] More applications of ideals can be seen in [2 3]We continue in this direction and introduce 119868-convergenceof generalized sequences with respect to Musielak-Orliczfunction
A familyI sub 2
119883 of subsets of a nonempty set119883 is said tobe an ideal in119883 if
(1) 120601 isin I(2) 119860 119861 isin I imply 119860 cup 119861 isin I(3) 119860 isin I 119861 sub 119860 imply 119861 isin I
while an admissible ideal I of 119883 further satisfies 119909 isin Ifor each 119909 isin 119883 see [1] A sequence (119909119899)119899isinN in 119883 is said to be119868-convergent to 119909 isin 119883 If for each 120598 gt 0 the set 119860(120598) = 119899 isinN 119909119899minus119909 ge 120598 belongs toI see [1] For more details aboutideal convergent sequence spaces see [4ndash10] and referencestherein
Mursaleen and Noman [11] introduced the notion of 120582-convergent and 120582-bounded sequences as follows
The sequence 119909 = (119909119896) isin 119908 is 120582-convergent to the number 119871called the 120582-limit of 119909 if Λ119898(119909) rarr 119871 as119898 rarr infin where
Λ119898 (119909) =
1
120582119898
119898
sum
119896=1
(120582119896 minus 120582119896minus1) 119909119896 (2)
The sequence 119909=(119909119896)isin119908 is 120582-bounded if sup119898|Λ119898(119909)| lt
infin It is well known [11] that if lim119898 119909119898 = 119886 in the ordinarysense of convergence then
lim119898(
1
120582119898
(
119898
sum
119896=1
(120582119896 minus 120582119896minus1)1003816100381610038161003816
119909119896 minus 1198861003816100381610038161003816
)) = 0 (3)
This implies that
lim119898
1003816100381610038161003816
Λ119898 (119909) minus 1198861003816100381610038161003816
which yields that lim119898Λ119898(119909) = 119886 and hence 119909 = (119909119896) isin 119908 is120582-convergent to 119886
Let 119883 be a linear metric space A function 119901 119883 rarr R iscalled paranorm if
(1) 119901(119909) ge 0 for all 119909 isin 119883(2) 119901(minus119909) = 119901(119909) for all 119909 isin 119883(3) 119901(119909 + 119910) le 119901(119909) + 119901(119910) for all 119909 119910 isin 119883
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 134534 6 pageshttpdxdoiorg1011552014134534
2 The Scientific World Journal
(4) if (120582119899) is a sequence of scalars with 120582119899 rarr 120582 as 119899 rarrinfin and (119909119899) is a sequence of vectors with 119901(119909119899minus119909) rarr0 as 119899 rarr infin then 119901(120582119899119909119899 minus 120582119909) rarr 0 as 119899 rarr infin
A paranorm 119901 for which 119901(119909) = 0 implies that 119909 = 0
is called total paranorm and the pair (119883 119901) is called a totalparanormed space It is well known that the metric of anylinear metric space is given by some total paranorm (see[12Theorem 1042 P-183]) For more details about sequencespaces see [13ndash15] and references therein
An Orlicz function119872 is a function which is continuousnondecreasing and convex with 119872(0) = 0 119872(119909) gt 0 for119909 gt 0 and119872(119909) rarr infin as 119909 rarr infin
Lindenstrauss and Tzafriri [16] used the idea of Orliczfunction to define the following sequence space Let 119908 be thespace of all real or complex sequences 119909 = (119909119896) Then
ℓ119872 = 119909 isin 119908
infin
sum
119896=1
119872(
1003816100381610038161003816
119909119896
1003816100381610038161003816
120588
) lt infin (5)
which is called an Orlicz sequence space The space ℓ119872 is aBanach space with the norm
119909 = inf 120588 gt 0 infin
sum
119896=1
119872(
1003816100381610038161003816
119909119896
1003816100381610038161003816
120588
) le 1 (6)
It is shown in [16] that every Orlicz sequence space ℓ119872contains a subspace isomorphic to ℓ119901 (119901 ge 1) The Δ 2-condition is equivalent to 119872(119871119909) le 119896119871119872(119909) for all valuesof 119909 ge 0 and for 119871 gt 1
A sequence M = (119872119896) of Orlicz function is called aMusielak-Orlicz function see [17 18] A sequenceN = (119873119896)
defined by
119873119896 (V) = sup |V| 119906 minus (119872119896) 119906 ge 0 119896 = 1 2 (7)
is called the complementary function of a Musielak-Orliczfunction M For a given Musielak-Orlicz function M theMusielak-Orlicz sequence space 119905M and its subspace ℎM aredefined as follows
119905M = 119909 isin 119908 119868M (119888119909) lt infin for some 119888 gt 0
ℎM = 119909 isin 119908 119868M (119888119909) lt infin for all 119888 gt 0 (8)
We consider 119905M equipped with the Luxemburg norm
119909 = inf 119896 gt 0 119868M (119909
119896
) le 1 (10)
or equipped with the Orlicz norm
119909
0= inf 1
119896
(1 + 119868M (119896119909)) 119896 gt 0 (11)
Let M = (119872119896) be a Musielak-Orlicz function and let119901 = (119901119896) be a bounded sequence of positive real numbersWe define the following sequence spaces
119888
119868(M Λ 119901)
= 119909 = (119909119896) isin 119908 119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
= 0
for some 119871 and 120588 gt 0
119888
119868
0(M Λ 119901)
= 119909 = (119909119896) isin 119908 119868 minus lim119896
The following inequality will be used throughout thepaper If 0 le 119901119896 le sup119901119896 = 119867119863 = max(1 2119867minus1) then
1003816100381610038161003816
119886119896 + 119887119896
1003816100381610038161003816
119901119896le 119863
1003816100381610038161003816
119886119896
1003816100381610038161003816
119901119896+
1003816100381610038161003816
119861119896
1003816100381610038161003816
119901119896 (15)
for all 119896 and 119886119896 119887119896 isin C Also |119886|119901119896 le max(1 |119886|119867) for all119886 isin C
The main aim of this paper is to study some ideal conver-gent sequence spaces defined by a Musielak-Orlicz functionM = (119872119896) We also make an effort to study some topologicalproperties and prove some inclusion relations between thesespaces
2 Main Results
Theorem 1 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen the spaces 119888119868(M Λ 119901) 119888119868
0(M Λ 119901) 119898119868(M Λ 119901) and
119898
119868
0(M Λ 119901) are linear
Proof Let 119909 119910 isin 119888119868(M Λ 119901) and let 120572 120573 be scalars Thenthere exist positive numbers 1205881 and 1205882 such that
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
= 0 for some 1198711 isin C
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
= 0 for some 1198712 isin C
(16)
For a given 120598 gt 0 we have
1198631 = 119896 isin N 119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
1198632 = 119896 isin N 119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
(17)
Let 1205883 = max2|120572|1205881 2|120573|1205882 Since M = (119872119896) is non-decreasing convex function so by using inequality (15) wehave
lim119896
119872119896(
|Λ 119896((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))|
1205883
)
119901119896
le lim119896
119872119896(
|120572|
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205883
+
1003816100381610038161003816
120573
1003816100381610038161003816
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205883
)
119901119896
le lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
+ lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
(18)
Now by (17) we have
119896 isin N lim119896
119872119896(
1003816100381610038161003816
Λ 119896 ((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))1003816100381610038161003816
1205883
)
119901119896
gt 120598 sub 1198631 cup 1198632
(19)
Therefore 120572119909 + 120573119910 isin 119888
119868(M Λ 119901) Hence 119888119868(M Λ 119901)
is a linear space Similarly we can prove that 1198881198680(M Λ 119901)
119898
119868(M Λ 119901) and119898119868
0(M Λ 119901) are linear spaces
Theorem 2 Let M = (119872119896) be a Musielak-Orlicz functionThen
119888
119868
0(M Λ 119901) sub 119888
119868(M Λ 119901) sub 119897infin (M Λ 119901) (20)
Proof Let 119909 isin 119888119868(M Λ 119901) Then there exist 119871 isin C and 120588 gt 0such that
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
= 0 (21)
We have
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
2120588
)
119901119896
le
1
2
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
+119872119896
1
2
(
|119871|
120588
)
119901119896
(22)
Taking supremum over 119896 on both sides we get 119909 isin
119897infin(M Λ 119901) The inclusion 1198881198680(M Λ 119901) sub 119888
119868(M Λ 119901) is
obvious Thus
119888
119868
0(M Λ 119901) sub 119888
119868(M Λ 119901) sub 119897infin (M Λ 119901) (23)
This completes the proof of the theorem
Theorem 3 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen 119897infin(M Λ 119901) is a paranormed space with paranormdefined by
119892 (119909) = inf 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
120588
)
119901119896
le 1 (24)
Proof It is clear that 119892(119909) = 119892(minus119909) Since119872119896(0) = 0 we get119892(0) = 0 Let us take 119909 119910 isin 119897infin (M Λ 119901) Let
119861 (119909) = 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
120588
)
119901119896
le 1
119861 (119910) = 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910)1003816100381610038161003816
120588
)
119901119896
le 1
(25)
4 The Scientific World Journal
Let 1205881 isin 119861(119909) and 1205882 isin 119861(119910) If 120588 = 1205881 + 1205882 then we have
Let 120590119904 rarr 120590 where 120590 120590119904 isin C and let 119892(119909119904 minus 119909) rarr 0 as119904 rarr infin We have to show that 119892(120590119904119909119904minus120590119909) rarr 0 as 119904 rarr infinLet
119861 (119909
119904) = 120588119904 gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904)
1003816100381610038161003816
120588119904
)
119901119896
le 1
119861 (119909
119904minus 119909) = 120588
1015840
119904gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904minus 119909)
1003816100381610038161003816
120588
1015840119904
)
119901119896
le 1
(28)
If 120588119904 isin 119861(119909119904) and 1205881015840
sequence algebra Similarly we can prove that 119888119868(M Λ 119901) isa sequence algebra
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[2] P Das P Kostyrko W Wilczynski and P Malik ldquo119868 and 119868lowast-convergence of double sequencesrdquo Mathematica Slovaca vol58 no 5 pp 605ndash620 2008
[3] P Das and P Malik ldquoOn the statistical and 119868 variation of doublesequencesrdquo Real Analysis Exchange vol 33 no 2 pp 351ndash3632008
[4] V Kumar ldquoOn 119868 and 119868lowast-convergence of double sequencesrdquoMathematical Communications vol 12 no 2 pp 171ndash181 2007
[5] M Mursaleen and A Alotaibi ldquoOn 119868-convergence in random2-normed spacesrdquoMathematica Slovaca vol 61 no 6 pp 933ndash940 2011
[6] M Mursaleen S A Mohiuddine and O H H Edely ldquoOn theideal convergence of double sequences in intuitionistic fuzzynormed spacesrdquo Computers amp Mathematics with Applicationsvol 59 no 2 pp 603ndash611 2010
[7] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence ofdouble sequences in probabilistic normed spacesrdquo Mathemati-cal Reports vol 12(62) no 4 pp 359ndash371 2010
[8] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012
[9] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2-normed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007
[10] B C Tripathy and B Hazarika ldquoSome 119868-convergent sequencespaces defined by Orlicz functionsrdquo Acta Mathematicae Appli-catae Sinica vol 27 no 1 pp 149ndash154 2011
[11] M Mursaleen and A K Noman ldquoOn some new sequencespaces of non-absolute type related to the spaces ℓ119901 and ℓinfin IrdquoFilomat vol 25 no 2 pp 33ndash51 2011
[12] A Wilansky Summability through Functional Analysis vol 85ofNorth-HollandMathematics Studies North-Holland Publish-ing Amsterdam The Netherlands 1984
6 The Scientific World Journal
[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011
[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012
[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012
[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971
[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989
[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983
(4) if (120582119899) is a sequence of scalars with 120582119899 rarr 120582 as 119899 rarrinfin and (119909119899) is a sequence of vectors with 119901(119909119899minus119909) rarr0 as 119899 rarr infin then 119901(120582119899119909119899 minus 120582119909) rarr 0 as 119899 rarr infin
A paranorm 119901 for which 119901(119909) = 0 implies that 119909 = 0
is called total paranorm and the pair (119883 119901) is called a totalparanormed space It is well known that the metric of anylinear metric space is given by some total paranorm (see[12Theorem 1042 P-183]) For more details about sequencespaces see [13ndash15] and references therein
An Orlicz function119872 is a function which is continuousnondecreasing and convex with 119872(0) = 0 119872(119909) gt 0 for119909 gt 0 and119872(119909) rarr infin as 119909 rarr infin
Lindenstrauss and Tzafriri [16] used the idea of Orliczfunction to define the following sequence space Let 119908 be thespace of all real or complex sequences 119909 = (119909119896) Then
ℓ119872 = 119909 isin 119908
infin
sum
119896=1
119872(
1003816100381610038161003816
119909119896
1003816100381610038161003816
120588
) lt infin (5)
which is called an Orlicz sequence space The space ℓ119872 is aBanach space with the norm
119909 = inf 120588 gt 0 infin
sum
119896=1
119872(
1003816100381610038161003816
119909119896
1003816100381610038161003816
120588
) le 1 (6)
It is shown in [16] that every Orlicz sequence space ℓ119872contains a subspace isomorphic to ℓ119901 (119901 ge 1) The Δ 2-condition is equivalent to 119872(119871119909) le 119896119871119872(119909) for all valuesof 119909 ge 0 and for 119871 gt 1
A sequence M = (119872119896) of Orlicz function is called aMusielak-Orlicz function see [17 18] A sequenceN = (119873119896)
defined by
119873119896 (V) = sup |V| 119906 minus (119872119896) 119906 ge 0 119896 = 1 2 (7)
is called the complementary function of a Musielak-Orliczfunction M For a given Musielak-Orlicz function M theMusielak-Orlicz sequence space 119905M and its subspace ℎM aredefined as follows
119905M = 119909 isin 119908 119868M (119888119909) lt infin for some 119888 gt 0
ℎM = 119909 isin 119908 119868M (119888119909) lt infin for all 119888 gt 0 (8)
We consider 119905M equipped with the Luxemburg norm
119909 = inf 119896 gt 0 119868M (119909
119896
) le 1 (10)
or equipped with the Orlicz norm
119909
0= inf 1
119896
(1 + 119868M (119896119909)) 119896 gt 0 (11)
Let M = (119872119896) be a Musielak-Orlicz function and let119901 = (119901119896) be a bounded sequence of positive real numbersWe define the following sequence spaces
119888
119868(M Λ 119901)
= 119909 = (119909119896) isin 119908 119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
= 0
for some 119871 and 120588 gt 0
119888
119868
0(M Λ 119901)
= 119909 = (119909119896) isin 119908 119868 minus lim119896
The following inequality will be used throughout thepaper If 0 le 119901119896 le sup119901119896 = 119867119863 = max(1 2119867minus1) then
1003816100381610038161003816
119886119896 + 119887119896
1003816100381610038161003816
119901119896le 119863
1003816100381610038161003816
119886119896
1003816100381610038161003816
119901119896+
1003816100381610038161003816
119861119896
1003816100381610038161003816
119901119896 (15)
for all 119896 and 119886119896 119887119896 isin C Also |119886|119901119896 le max(1 |119886|119867) for all119886 isin C
The main aim of this paper is to study some ideal conver-gent sequence spaces defined by a Musielak-Orlicz functionM = (119872119896) We also make an effort to study some topologicalproperties and prove some inclusion relations between thesespaces
2 Main Results
Theorem 1 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen the spaces 119888119868(M Λ 119901) 119888119868
0(M Λ 119901) 119898119868(M Λ 119901) and
119898
119868
0(M Λ 119901) are linear
Proof Let 119909 119910 isin 119888119868(M Λ 119901) and let 120572 120573 be scalars Thenthere exist positive numbers 1205881 and 1205882 such that
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
= 0 for some 1198711 isin C
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
= 0 for some 1198712 isin C
(16)
For a given 120598 gt 0 we have
1198631 = 119896 isin N 119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
1198632 = 119896 isin N 119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
(17)
Let 1205883 = max2|120572|1205881 2|120573|1205882 Since M = (119872119896) is non-decreasing convex function so by using inequality (15) wehave
lim119896
119872119896(
|Λ 119896((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))|
1205883
)
119901119896
le lim119896
119872119896(
|120572|
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205883
+
1003816100381610038161003816
120573
1003816100381610038161003816
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205883
)
119901119896
le lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
+ lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
(18)
Now by (17) we have
119896 isin N lim119896
119872119896(
1003816100381610038161003816
Λ 119896 ((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))1003816100381610038161003816
1205883
)
119901119896
gt 120598 sub 1198631 cup 1198632
(19)
Therefore 120572119909 + 120573119910 isin 119888
119868(M Λ 119901) Hence 119888119868(M Λ 119901)
is a linear space Similarly we can prove that 1198881198680(M Λ 119901)
119898
119868(M Λ 119901) and119898119868
0(M Λ 119901) are linear spaces
Theorem 2 Let M = (119872119896) be a Musielak-Orlicz functionThen
119888
119868
0(M Λ 119901) sub 119888
119868(M Λ 119901) sub 119897infin (M Λ 119901) (20)
Proof Let 119909 isin 119888119868(M Λ 119901) Then there exist 119871 isin C and 120588 gt 0such that
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
= 0 (21)
We have
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
2120588
)
119901119896
le
1
2
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
+119872119896
1
2
(
|119871|
120588
)
119901119896
(22)
Taking supremum over 119896 on both sides we get 119909 isin
119897infin(M Λ 119901) The inclusion 1198881198680(M Λ 119901) sub 119888
119868(M Λ 119901) is
obvious Thus
119888
119868
0(M Λ 119901) sub 119888
119868(M Λ 119901) sub 119897infin (M Λ 119901) (23)
This completes the proof of the theorem
Theorem 3 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen 119897infin(M Λ 119901) is a paranormed space with paranormdefined by
119892 (119909) = inf 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
120588
)
119901119896
le 1 (24)
Proof It is clear that 119892(119909) = 119892(minus119909) Since119872119896(0) = 0 we get119892(0) = 0 Let us take 119909 119910 isin 119897infin (M Λ 119901) Let
119861 (119909) = 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
120588
)
119901119896
le 1
119861 (119910) = 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910)1003816100381610038161003816
120588
)
119901119896
le 1
(25)
4 The Scientific World Journal
Let 1205881 isin 119861(119909) and 1205882 isin 119861(119910) If 120588 = 1205881 + 1205882 then we have
Let 120590119904 rarr 120590 where 120590 120590119904 isin C and let 119892(119909119904 minus 119909) rarr 0 as119904 rarr infin We have to show that 119892(120590119904119909119904minus120590119909) rarr 0 as 119904 rarr infinLet
119861 (119909
119904) = 120588119904 gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904)
1003816100381610038161003816
120588119904
)
119901119896
le 1
119861 (119909
119904minus 119909) = 120588
1015840
119904gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904minus 119909)
1003816100381610038161003816
120588
1015840119904
)
119901119896
le 1
(28)
If 120588119904 isin 119861(119909119904) and 1205881015840
sequence algebra Similarly we can prove that 119888119868(M Λ 119901) isa sequence algebra
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[2] P Das P Kostyrko W Wilczynski and P Malik ldquo119868 and 119868lowast-convergence of double sequencesrdquo Mathematica Slovaca vol58 no 5 pp 605ndash620 2008
[3] P Das and P Malik ldquoOn the statistical and 119868 variation of doublesequencesrdquo Real Analysis Exchange vol 33 no 2 pp 351ndash3632008
[4] V Kumar ldquoOn 119868 and 119868lowast-convergence of double sequencesrdquoMathematical Communications vol 12 no 2 pp 171ndash181 2007
[5] M Mursaleen and A Alotaibi ldquoOn 119868-convergence in random2-normed spacesrdquoMathematica Slovaca vol 61 no 6 pp 933ndash940 2011
[6] M Mursaleen S A Mohiuddine and O H H Edely ldquoOn theideal convergence of double sequences in intuitionistic fuzzynormed spacesrdquo Computers amp Mathematics with Applicationsvol 59 no 2 pp 603ndash611 2010
[7] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence ofdouble sequences in probabilistic normed spacesrdquo Mathemati-cal Reports vol 12(62) no 4 pp 359ndash371 2010
[8] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012
[9] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2-normed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007
[10] B C Tripathy and B Hazarika ldquoSome 119868-convergent sequencespaces defined by Orlicz functionsrdquo Acta Mathematicae Appli-catae Sinica vol 27 no 1 pp 149ndash154 2011
[11] M Mursaleen and A K Noman ldquoOn some new sequencespaces of non-absolute type related to the spaces ℓ119901 and ℓinfin IrdquoFilomat vol 25 no 2 pp 33ndash51 2011
[12] A Wilansky Summability through Functional Analysis vol 85ofNorth-HollandMathematics Studies North-Holland Publish-ing Amsterdam The Netherlands 1984
6 The Scientific World Journal
[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011
[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012
[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012
[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971
[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989
[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983
The following inequality will be used throughout thepaper If 0 le 119901119896 le sup119901119896 = 119867119863 = max(1 2119867minus1) then
1003816100381610038161003816
119886119896 + 119887119896
1003816100381610038161003816
119901119896le 119863
1003816100381610038161003816
119886119896
1003816100381610038161003816
119901119896+
1003816100381610038161003816
119861119896
1003816100381610038161003816
119901119896 (15)
for all 119896 and 119886119896 119887119896 isin C Also |119886|119901119896 le max(1 |119886|119867) for all119886 isin C
The main aim of this paper is to study some ideal conver-gent sequence spaces defined by a Musielak-Orlicz functionM = (119872119896) We also make an effort to study some topologicalproperties and prove some inclusion relations between thesespaces
2 Main Results
Theorem 1 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen the spaces 119888119868(M Λ 119901) 119888119868
0(M Λ 119901) 119898119868(M Λ 119901) and
119898
119868
0(M Λ 119901) are linear
Proof Let 119909 119910 isin 119888119868(M Λ 119901) and let 120572 120573 be scalars Thenthere exist positive numbers 1205881 and 1205882 such that
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
= 0 for some 1198711 isin C
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
= 0 for some 1198712 isin C
(16)
For a given 120598 gt 0 we have
1198631 = 119896 isin N 119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
1198632 = 119896 isin N 119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
(17)
Let 1205883 = max2|120572|1205881 2|120573|1205882 Since M = (119872119896) is non-decreasing convex function so by using inequality (15) wehave
lim119896
119872119896(
|Λ 119896((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))|
1205883
)
119901119896
le lim119896
119872119896(
|120572|
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205883
+
1003816100381610038161003816
120573
1003816100381610038161003816
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205883
)
119901119896
le lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711
1003816100381610038161003816
1205881
)
119901119896
+ lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910) minus 1198712
1003816100381610038161003816
1205882
)
119901119896
(18)
Now by (17) we have
119896 isin N lim119896
119872119896(
1003816100381610038161003816
Λ 119896 ((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))1003816100381610038161003816
1205883
)
119901119896
gt 120598 sub 1198631 cup 1198632
(19)
Therefore 120572119909 + 120573119910 isin 119888
119868(M Λ 119901) Hence 119888119868(M Λ 119901)
is a linear space Similarly we can prove that 1198881198680(M Λ 119901)
119898
119868(M Λ 119901) and119898119868
0(M Λ 119901) are linear spaces
Theorem 2 Let M = (119872119896) be a Musielak-Orlicz functionThen
119888
119868
0(M Λ 119901) sub 119888
119868(M Λ 119901) sub 119897infin (M Λ 119901) (20)
Proof Let 119909 isin 119888119868(M Λ 119901) Then there exist 119871 isin C and 120588 gt 0such that
119868 minus lim119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
= 0 (21)
We have
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
2120588
)
119901119896
le
1
2
119872119896(
1003816100381610038161003816
Λ 119896 (119909) minus 1198711003816100381610038161003816
120588
)
119901119896
+119872119896
1
2
(
|119871|
120588
)
119901119896
(22)
Taking supremum over 119896 on both sides we get 119909 isin
119897infin(M Λ 119901) The inclusion 1198881198680(M Λ 119901) sub 119888
119868(M Λ 119901) is
obvious Thus
119888
119868
0(M Λ 119901) sub 119888
119868(M Λ 119901) sub 119897infin (M Λ 119901) (23)
This completes the proof of the theorem
Theorem 3 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen 119897infin(M Λ 119901) is a paranormed space with paranormdefined by
119892 (119909) = inf 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
120588
)
119901119896
le 1 (24)
Proof It is clear that 119892(119909) = 119892(minus119909) Since119872119896(0) = 0 we get119892(0) = 0 Let us take 119909 119910 isin 119897infin (M Λ 119901) Let
119861 (119909) = 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909)1003816100381610038161003816
120588
)
119901119896
le 1
119861 (119910) = 120588 gt 0 sup119896
119872119896(
1003816100381610038161003816
Λ 119896 (119910)1003816100381610038161003816
120588
)
119901119896
le 1
(25)
4 The Scientific World Journal
Let 1205881 isin 119861(119909) and 1205882 isin 119861(119910) If 120588 = 1205881 + 1205882 then we have
Let 120590119904 rarr 120590 where 120590 120590119904 isin C and let 119892(119909119904 minus 119909) rarr 0 as119904 rarr infin We have to show that 119892(120590119904119909119904minus120590119909) rarr 0 as 119904 rarr infinLet
119861 (119909
119904) = 120588119904 gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904)
1003816100381610038161003816
120588119904
)
119901119896
le 1
119861 (119909
119904minus 119909) = 120588
1015840
119904gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904minus 119909)
1003816100381610038161003816
120588
1015840119904
)
119901119896
le 1
(28)
If 120588119904 isin 119861(119909119904) and 1205881015840
sequence algebra Similarly we can prove that 119888119868(M Λ 119901) isa sequence algebra
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[2] P Das P Kostyrko W Wilczynski and P Malik ldquo119868 and 119868lowast-convergence of double sequencesrdquo Mathematica Slovaca vol58 no 5 pp 605ndash620 2008
[3] P Das and P Malik ldquoOn the statistical and 119868 variation of doublesequencesrdquo Real Analysis Exchange vol 33 no 2 pp 351ndash3632008
[4] V Kumar ldquoOn 119868 and 119868lowast-convergence of double sequencesrdquoMathematical Communications vol 12 no 2 pp 171ndash181 2007
[5] M Mursaleen and A Alotaibi ldquoOn 119868-convergence in random2-normed spacesrdquoMathematica Slovaca vol 61 no 6 pp 933ndash940 2011
[6] M Mursaleen S A Mohiuddine and O H H Edely ldquoOn theideal convergence of double sequences in intuitionistic fuzzynormed spacesrdquo Computers amp Mathematics with Applicationsvol 59 no 2 pp 603ndash611 2010
[7] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence ofdouble sequences in probabilistic normed spacesrdquo Mathemati-cal Reports vol 12(62) no 4 pp 359ndash371 2010
[8] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012
[9] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2-normed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007
[10] B C Tripathy and B Hazarika ldquoSome 119868-convergent sequencespaces defined by Orlicz functionsrdquo Acta Mathematicae Appli-catae Sinica vol 27 no 1 pp 149ndash154 2011
[11] M Mursaleen and A K Noman ldquoOn some new sequencespaces of non-absolute type related to the spaces ℓ119901 and ℓinfin IrdquoFilomat vol 25 no 2 pp 33ndash51 2011
[12] A Wilansky Summability through Functional Analysis vol 85ofNorth-HollandMathematics Studies North-Holland Publish-ing Amsterdam The Netherlands 1984
6 The Scientific World Journal
[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011
[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012
[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012
[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971
[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989
[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983
Let 120590119904 rarr 120590 where 120590 120590119904 isin C and let 119892(119909119904 minus 119909) rarr 0 as119904 rarr infin We have to show that 119892(120590119904119909119904minus120590119909) rarr 0 as 119904 rarr infinLet
119861 (119909
119904) = 120588119904 gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904)
1003816100381610038161003816
120588119904
)
119901119896
le 1
119861 (119909
119904minus 119909) = 120588
1015840
119904gt 0 sup
119896
119872119896(
1003816100381610038161003816
Λ 119896 (119909119904minus 119909)
1003816100381610038161003816
120588
1015840119904
)
119901119896
le 1
(28)
If 120588119904 isin 119861(119909119904) and 1205881015840
sequence algebra Similarly we can prove that 119888119868(M Λ 119901) isa sequence algebra
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[2] P Das P Kostyrko W Wilczynski and P Malik ldquo119868 and 119868lowast-convergence of double sequencesrdquo Mathematica Slovaca vol58 no 5 pp 605ndash620 2008
[3] P Das and P Malik ldquoOn the statistical and 119868 variation of doublesequencesrdquo Real Analysis Exchange vol 33 no 2 pp 351ndash3632008
[4] V Kumar ldquoOn 119868 and 119868lowast-convergence of double sequencesrdquoMathematical Communications vol 12 no 2 pp 171ndash181 2007
[5] M Mursaleen and A Alotaibi ldquoOn 119868-convergence in random2-normed spacesrdquoMathematica Slovaca vol 61 no 6 pp 933ndash940 2011
[6] M Mursaleen S A Mohiuddine and O H H Edely ldquoOn theideal convergence of double sequences in intuitionistic fuzzynormed spacesrdquo Computers amp Mathematics with Applicationsvol 59 no 2 pp 603ndash611 2010
[7] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence ofdouble sequences in probabilistic normed spacesrdquo Mathemati-cal Reports vol 12(62) no 4 pp 359ndash371 2010
[8] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012
[9] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2-normed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007
[10] B C Tripathy and B Hazarika ldquoSome 119868-convergent sequencespaces defined by Orlicz functionsrdquo Acta Mathematicae Appli-catae Sinica vol 27 no 1 pp 149ndash154 2011
[11] M Mursaleen and A K Noman ldquoOn some new sequencespaces of non-absolute type related to the spaces ℓ119901 and ℓinfin IrdquoFilomat vol 25 no 2 pp 33ndash51 2011
[12] A Wilansky Summability through Functional Analysis vol 85ofNorth-HollandMathematics Studies North-Holland Publish-ing Amsterdam The Netherlands 1984
6 The Scientific World Journal
[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011
[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012
[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012
[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971
[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989
[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983
sequence algebra Similarly we can prove that 119888119868(M Λ 119901) isa sequence algebra
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[2] P Das P Kostyrko W Wilczynski and P Malik ldquo119868 and 119868lowast-convergence of double sequencesrdquo Mathematica Slovaca vol58 no 5 pp 605ndash620 2008
[3] P Das and P Malik ldquoOn the statistical and 119868 variation of doublesequencesrdquo Real Analysis Exchange vol 33 no 2 pp 351ndash3632008
[4] V Kumar ldquoOn 119868 and 119868lowast-convergence of double sequencesrdquoMathematical Communications vol 12 no 2 pp 171ndash181 2007
[5] M Mursaleen and A Alotaibi ldquoOn 119868-convergence in random2-normed spacesrdquoMathematica Slovaca vol 61 no 6 pp 933ndash940 2011
[6] M Mursaleen S A Mohiuddine and O H H Edely ldquoOn theideal convergence of double sequences in intuitionistic fuzzynormed spacesrdquo Computers amp Mathematics with Applicationsvol 59 no 2 pp 603ndash611 2010
[7] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence ofdouble sequences in probabilistic normed spacesrdquo Mathemati-cal Reports vol 12(62) no 4 pp 359ndash371 2010
[8] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012
[9] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2-normed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007
[10] B C Tripathy and B Hazarika ldquoSome 119868-convergent sequencespaces defined by Orlicz functionsrdquo Acta Mathematicae Appli-catae Sinica vol 27 no 1 pp 149ndash154 2011
[11] M Mursaleen and A K Noman ldquoOn some new sequencespaces of non-absolute type related to the spaces ℓ119901 and ℓinfin IrdquoFilomat vol 25 no 2 pp 33ndash51 2011
[12] A Wilansky Summability through Functional Analysis vol 85ofNorth-HollandMathematics Studies North-Holland Publish-ing Amsterdam The Netherlands 1984
6 The Scientific World Journal
[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011
[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012
[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012
[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971
[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989
[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983
[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011
[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012
[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012
[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971
[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989
[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983