# Complex Convexity of Orlicz Modular Sequence Spaces

Authors: Lili Chen; Deyun Chen; Yang Jiang
Journal: Journal of Function Spaces
 (2016)

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## Abstract

The concepts of complex extreme points, complex strongly extreme points, complex strict convexity, and complex midpoint locally uniform convexity in general modular spaces are introduced. Then we prove that, for any Orlicz modular sequence spacelΦ,ρ, lΦ,ρ is complex midpoint locally uniformly convex. As a corollary, lΦ,ρ is also complex strictly convex.


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## Body

## 1. Introduction

In 1967, the notions of complex extreme points and complex strict convexity have been introduced by Thorp and Whitley [1]. They proved that the strong maximum modulus theorem for analytic functions with values in a complex Banach space X holds true whenever each point of the unit sphere of X is a complex extreme point. In 1975, Globevnik further introduced the notions of complex strict and uniform convexity of complex normed spaces and proved that the complex space L1 is complex uniformly convex (see [2]). Davis et al. in [3] investigated the complex convexity of quasi-normed linear spaces. Dowling et al. in [4] studied the complex convexity of Lebesgue-Bochner function spaces. Blasco and Pavlović in [5] obtained sufficient and necessary conditions for a complex Banach space X which is p-uniformly PL-convex. Choi et al. in [6] obtained criteria for complex extreme points, complex rotundity, and complex uniform convexity in Orlicz-Lorentz spaces. Hudzik and Narloch in [7] considered relationships between monotonicity and complex rotundity; for instance, a point f of the complexification EC of a real Köthe space E is a complex extreme point if and only if |f| is a point of upper monotonicity in E. Lee in [8, 9] continued to study relationships between monotonicity and complex convexity in Banach lattices and Quasi-Banach lattices, respectively. Czerwińska and Kamińska in [10] discussed the complex rotundity and midpoint local uniform convexity in symmetric spaces of measurable operators. Recently, Czerwińska and Parrish in [11] characterized complex extreme points in Marcinkiewicz spaces.In this paper, we introduce the concepts of complex extreme points, complex strongly extreme points, complex strict convexity, and complex midpoint locally uniform convexity in general modular spaces. Then we prove that, for any Orlicz modular sequence spacelΦ,ρ, lΦ,ρ is complex midpoint locally uniformly convex. As a corollary, lΦ,ρ is also complex strictly convex.Before starting with our results, we need to recall some basic concepts and facts of the theory of modular spaces and Orlicz spaces.LetX be a vector space over the complex field C. A functional ρ:X→[0,∞] is called a modular provided that, for any f,g∈X,(a)
ρ(f)=0 if and only if f=0;(b)
ρ(αf)=ρ(f) for any α∈C with |α|=1;(c)
ρ(αf+βg)≤ρ(f)+ρ(g) for any α,β≥0 with α+β=1; we can replace (c) by the following;(c′)
ρ(αf+βg)≤αρ(f)+βρ(g) for any α,β≥0 with α+β=1; in this case the modular ρ is said to be a convex modular. A modular space Xρ is defined by(1)Xρ=f∈X:ρλf⟶0asλ⟶0.LetXρ be a modular space; then (2)BXρ=x∈Xρ:ρx≤1,SXρ=x∈Xρ:ρx=1denote the closed unit ball and the unit sphere of Xρ, respectively. In the sequel N, R, and C denote the set of natural numbers, the set of real numbers, and the set of complex numbers, respectively. Let i be the complex number satisfying i2=-1.A mapΦ:R→[0,∞] is said to be an Orlicz function if Φ is vanishing at zero, even, convex, and continuous and satisfies limu→0(Φu/u)=0 and limu→∞(Φu/u)=∞. For every Orlicz function Φ, its complementary function Ψ:R→[0,∞] is defined by the formula (3)Ψv=sup⁡uv-Φu:u≥0,and the complementary function Ψ is also an Orlicz function.Define(4)lc=x=xjj=1∞:xj∈C,j=1,2,….For any x∈lc, let supp⁡(x)={j∈N:x(j)≠0} denote the support set of x. For a given N-function Φ, we define on lc a convex modular by (5)ρΦx=∑j=1∞Φxj,∀x=xjj=1∞∈lc,which is said to be an Orlicz modular. The modular space induced by an Orlicz modular is said to be the Orlicz sequence space lΦ. Indeed, Orlicz sequence spaces and their kinds of generalizations belong to modular spaces. For the sake of simplicity, we denote lΦ,ρ=(lΦ,ρΦ) and lΦ=(lΦ,·).For more details on Orlicz spaces we refer to [12–24], and for more details on modular spaces one can consult [25–33].


## 2. Main Results

In this section, we always assume thatρ is a convex modular. It is known that extreme points which are connected with strict convexity of the whole spaces are the most basic and important geometric points in geometric theory of spaces. In [1], Thorp and Whitley first introduced the concepts of complex extreme points and complex strict convexity when they studied the conditions under which the strong maximum modulus theorem for analytic functions always holds in a complex Banach space.Definition 1 (see [1]).
Let(X,·) be a Banach space. A point x∈S(X) is said to be a complex extreme point of B(X) if for every nonzero y∈X there holds sup|λ|≤1x+λy>1. A Banach space X is said to be complex strictly convex if every element of S(X) is a complex extreme point of B(X).In [12], we further studied the notions of complex strongly extreme points and complex midpoint locally uniform convexity in general complex spaces. For more details on notions of complex extreme points and complex strongly extreme points we refer to [12, 13].Definition 2 (see [12]).
Let(X,·) be a Banach space. A point x∈S(X) is said to be a complex strongly extreme point of B(X) if, for every ε>0, one has Δc(x,ε)>0, where(6)Δcx,ε=inf⁡1-λ:∃y∈Xs.t.x±yλ≤1,x±iyλ≤1,y≥ε.A Banach space X is said to be complex midpoint locally uniformly convex if every element of S(X) is a complex strongly extreme point of B(X).Now we generalize the notions of complex extreme points and complex strongly extreme points of Banach spaces to the modular spaces.Definition 3.
LetXρ be a modular space. A point x∈S(Xρ) is said to be a complex extreme point of B(Xρ) if, for any y∈Xρ with y≠0, there holds(7)supλ≤1ρx+λy>1.Xρ is said to be complex strictly convex if every element of S(Xρ) is a complex extreme point of B(Xρ).Definition 4.
LetXρ be a modular space. A point x∈S(Xρ) is said to be a complex strongly extreme point of B(Xρ) if Δc,ρ(x,ε)>0 for every ε>0, where(8)Δc,ρx,ε=inf⁡1-λ:∃y∈Xρs.t.ρx±yλ≤1λ,ρx±iyλ≤1λ,ρy≥εand λ∈C with 0<|λ|≤1. Xρ is said to be complex midpoint locally uniformly convex if every element of S(Xρ) is a complex strongly extreme point of B(Xρ).Proposition 5.
Letx∈S(Xρ), y∈Xρ, and m≥1; then the following conditions are equivalent:(i)
sup|λ|≤1ρ(x+λy)≤m for all λ∈C with |λ|≤1.(ii)
ρ(x±y)≤m and ρ(x±iy)≤m.Proof.
(i)⇒(ii) is trivial. For anyλ∈C with |λ|≤1, without loss of generality, there exist k1,k2∈[0,1] such that(9)x+λy=k1x+y+k2x+iy.It follows that(10)k1+k2=1,k1+ik2=λ.Consequently, (11)ρx+λy=ρk1x+y+k2x+iy≤k1ρx+y+k2ρx+iy≤m.Remark 6.
By Proposition5, it is easy to see that the inequality in Definition 3(12)supλ≤1⁡ρx+λy≤1is equivalent to the inequality maxλ=±1,±iρ(x+λy)≤1.Next, we give a new modulus which is associated with complex strongly extreme points.Definition 7.
LetXρ be a modular space and x∈S(Xρ). For each ε>0, one defines a function as follows:(13)δc,ρx,ε=inf⁡supλ≤1⁡ρx+λy-1:ρy≥ε.Observe that δc,ρ(x,ε)≥0 for all ε>0.Proposition 8.
LetXρ be a modular space. A point x∈S(Xρ) is a complex strongly extreme point of B(Xρ) if and only if δc,ρ(x,ε)>0 for all ε>0.Proof.
Necessity. Let x∈S(Xρ) be a complex strongly extreme point of B(Xρ). Assume that there exists ε0>0 such that δc,ρ(x,ε0)=0. By Definition 7, we can find a sequence {yn}⊆Xρ satisfying ρ(yn)≥ε0 and(14)supλ≤1⁡ρx+λyn⟶1asn⟶∞.Let αn=sup|λ|≤1ρ(x+λyn) and βn=1/αn. By (14) and noticing that 0<βn≤1 and βn→1 as n→∞, we obtain(15)ρx±βnyn≤1βn,ρx±iβnyn≤1βn,which implies Δc,ρ(x,ε0)=0, a contradiction.
Sufficiency. Suppose that x∈S(Xρ) is not a complex strongly extreme point of B(Xρ) and δc,ρ(x,ε)>0 for all ε>0. By Definition 4, then there exists ε0>0 such that Δc,ρ(x,ε0)=0. Hence, there are λn∈C with |λn|→1 and yn∈Xρ satisfying ρ(yn)≥ε0, such that(16)ρx±ynλn≤1λn,ρx±iynλn≤1λnfor each n∈N. Setting zn=yn/λn, we have (17)ρzn=ρynλn≥1λnρyn≥ε0λn≥ε0,ρx±zn≤1λn,ρx±izn≤1λn.In view of Proposition 5, we deduce that(18)supλ≤1ρx+λzn≤1λn.It follows that(19)δc,ρx,ε0≤supλ≤1ρx+λzn-1≤1λn-1⟶0asn⟶∞,which contradicts the fact that δc,ρ(x,ε)>0 for all ε>0.In Banach spaces, we have shown that ifx∈S(X) is a complex strongly extreme point of B(X), then x is a complex extreme point of B(X) (see [12]). We will prove that it is also true in modular spaces.Theorem 9.
LetXρ be a modular space. If x∈S(Xρ) is a complex strongly extreme point of B(Xρ), then x is a complex extreme point of B(Xρ).Proof.
Suppose thatx∈S(Xρ) is not a complex extreme point of the closed unit ball B(Xρ). Then there exists z∈Xρ∖{0} such that(20)supλ≤1ρx+λz≤1.Hence, we have(21)ρx±z≤1,ρx±iz≤1.Letting ε0=ρ(z)>0, then we obtain Δc,ρ(x,ε0)=0 which is a contradiction.The following result is an immediate corollary of the previous theorem.Corollary 10.
LetXρ be a modular space. If Xρ is complex midpoint locally uniformly convex, it is also complex strictly convex.Next, we will study complex convexity of Orlicz modular sequence spaces. Before starting with our results, let us recall a useful lemma.Lemma 11 (see [13]).
For anyε>0, there exists δ∈(0,1/2) such that if u,v∈C and (22)v≥ε8maxeu+ev,then (23)u≤1-2δ4Σeu+ev,where(24)maxe⁡u+ev=max⁡u+v,u-v,u+iv,u-iv,Σeu+ev=u+v+u-v+u+iv+u-iv.In [34], we have given criteria for complex strict convexity and complex midpoint locally uniform convexity of Orlicz sequence spaces equipped with the p-Amemiya norm.Theorem 12 (see [34]).
Assume1≤p<∞; then the following are equivalent:(i)
lΦ,p is complex midpoint locally uniformly convex.(ii)
lΦ,p is complex strictly convex.(iii)
aΦ=0.Theorem 13 (see [34]).
Ifp=∞, then lΦ,p is complex strictly convex if and only if(i)
Φ∈Δ2,(ii)
aΦ=0.Theorem 14 (see [34]).
Ifp=∞, then lΦ,p is complex midpoint locally uniformly convex if and only if(i)
Φ∈Δ2,(ii)
aΦ=0,(iii)
for anyx∈lΦ,p, ε>0 there exists δ>0 such that, for every y∈lΦ,p∖{0}, there holds y(1)<ε/3, where y=y(1)+y(2), y(1)=Σj∈Ay(j)ξj, y(2)=Σj∉Ay(j)ξj, ξj:ξ(j)=1,  ξ(m)=0 for any m≠j, and(25)A=j∈N:Σe=±1,±ixj+eyj≤41+δxj.We will show that the Orlicz modular sequence spacelΦ,ρ is complex midpoint locally uniformly convex without any condition.Theorem 15.
LetlΦ,ρ be an Orlicz modular sequence space. Then lΦ,ρ is complex midpoint locally uniformly convex.Proof.
Suppose thatx∈S(lΦ,ρ) is not a complex strongly extreme point of the unit ball B(lΦ,ρ). By Definition 4, then there exists ε0>0 such that Δc,ρ(x,ε0)=0. In a similar way to Proposition 8, we can find a sequence {zn}⊆lΦ,ρ satisfying ρ(zn)≥ε0 and (26)supλ≤1ρΦx+λzn≤1λn.
For the aboveε0>0, by Lemma 11, there exists δ0∈(0,1/2), such that if u,v∈C and (27)v≥ε08maxeu+ev,then(28)u≤1-2δ04Σeu+ev.
For everyn∈N, let (29)An=j∈N:znj≥ε08maxexj+eznj,zn1:zn1j=znjj∉An,zn1j=0j∈An,zn2:zn2j=0j∉An,zn2j=znjj∈An.It is easy to see that zn=zn(1)+zn(2) for every n∈N, and (30)ρΦzn1=∑j∉AnΦznj≤∑j∉AnΦε08maxexj+eznj≤ε08∑j∉AnΦmaxexj+eznj≤ε08∑eρΦx+ezn≤ε02λn<3ε04for n large enough since λn→1 as n→∞. Consequently, we obtain(31)ρΦzn2>ε04,which shows that An≠∅. Furthermore, we have (32)1=ρΦx=∑j∈AnΦxj+∑j∉AnΦxj≤∑j∈AnΦ1-2δ04Σexj+eznj+∑j∉AnΦ14Σexj+eznj≤1-2δ0∑j∈AnΦ14Σexj+eznj+∑j∉AnΦ14Σexj+eznj=∑j=1∞Φ14Σexj+eznj-2δ0∑j∈AnΦ14Σexj+eznj≤14∑j=1∞ΦΣexj+eznj-2δ0∑j∈AnΦ14Σexj+eznj≤1λn-2δ0∑j∈AnΦ14Σexj+eznj.Notice that (33)∑j∈AnΦ14Σexj+eznj≥∑j∈AnΦznj=ρΦzn2>ε04.Hence, (34)1=ρΦx≤1λn-2δ0∑j∈AnΦ14Σexj+eznj<1λn-δ0ε02.In view of |λn|→1 as n→∞, letting n→∞, then we get a contradiction(35)1=ρΦx<1λn-δ0ε02<1,which completes the proof.Combining Corollary10 and Theorem 15 we obtain immediately the following result.Corollary 16.
LetlΦ,ρ be an Orlicz modular sequence space. Then lΦ,ρ is complex strictly convex.



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