# Comparison between Certain Equivalent Norms Regarding Some Familiar Properties Implying WFPP

Authors: Helga Fetter; Berta Gamboa de Buen
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

In a Banach space with a basis we define a similar norm to the
norm shown by Lin to makel1 into a space with FPP and make a comparative study of certain geometric properties such as the Opial property, WNS, and uniform nonsquareness of the original space and the space with the new norm.


---

## Body

## 1. Introduction

Dowling et al. in [1] defined a norm in l1 which was used by Lin [2] to exhibit an equivalent norm which makes l1 into a space with the fixed point property (FPP). A similar norm can be defined in every Banach space X with a basis. Since l1 with its usual norm does not have FPP, we asked ourselves if this norm in these spaces would also improve properties that imply the weak fixed point property (WFPP). We found out that in some instances it does, in some cases the original norm has better properties, and in some cases you cannot compare them. We give several examples to illustrate our assertions.


## 2. TheΓ Norm in a Banach Space

We start by giving the definition of the generalization of the norm used by Lin in a Banach space with a basis.Definition 2.1.
Let(X,∥·∥) be a Banach space with a basis {en}. Let x=∑i=1∞xiei∈X and Qn:X→X be the projection Qn(∑i=1∞xiei)=∑i=n∞xiei. The basis {en} is called premonotone if ∥Qnx∥≥∥Qn+1x∥ and monotone if ∥Pnx∥≤∥Pn+1x∥ where Pnx=(I-Qn+1)x for every x∈X and for every n∈ℕ.Definition 2.2.
Let(X,∥·∥) be a Banach space with a basis {en} and Γ={γn}⊂ℝ with 0<γn<γn+1 and limnγn=1. Let x=∑i=1∞xiei∈X. Then if
(2.1)|‖x‖|=sup⁡nγn‖Qnx‖,|∥·∥| is a norm in X which we will call Γ-norm.
Clearly(2.2)γ1‖x‖≤|‖x‖|≤(sup⁡n‖Qn‖)‖x‖.
Observe that, if{en} is a basis in (X,∥·∥), then it is always premonotone in (X,|∥·∥|) and, if {en} is monotone in (X,∥·∥), then it is also monotone in (X,|∥·∥|). Also observe that since for every x∈X we have that limnγn∥Qnx∥=0, there exists n0 such that |∥x∥|=γn0∥Qn0x∥.
Next we define the properties related to wfpp we are going to analyze. The definition of GGLD is not the original one found in [3], but an equivalent one found in [4].Definition 2.3.
LetY be a Banach space.(1)
Y has the Opial property if for every weakly null sequence {xn}⊂Y and for every x∈Y, x≠0,(2.3)lim⁡sup⁡n‖xn‖<lim⁡sup⁡n‖xn-x‖.(2)
IfY has a basis, it has the generalized Gossez-Lami Dozo property (GGLD) [4] if, for every weakly null normalized block basic sequence {yn}, we have that
(2.4)lim⁡nsup⁡i,j≥n‖yi-yj‖>1.(3)
A bounded sequence{yn}⊂Y is called diametral if
(2.5)lim⁡nd(yn+1,conv{yi}i=1n)=diam⁡{yn}n=1∞.Y has weak normal structure (WNS) if there is no weakly null diametral nonzero sequence in Y.(4)
The coefficientJ(Y), related to uniform nonsquareness, since J(Y)<2 if and only if Y is uniformly nonsquare, is given by
(2.6)J(Y)=sup⁡{min⁡(‖x+y‖,‖x-y‖):x,y∈BY}.(5)
The coefficientR(Y) [5] is defined by
(2.7)R(Y)=sup⁡{lim⁡inf⁡n‖xn+x‖:{x},{xn}⊂BY,xn⟶w0}.(6)
CoefficientsRW(a,Y) and MW(Y) [6] are defined as follows: for each a>0(2.8)RW(a,Y)=sup⁡{min⁡(lim⁡inf⁡n‖xn+x‖,lim⁡inf⁡n‖xn-x‖):‖x‖≤a,{xn}⊂BY,xn⟶w0},MW(Y)=sup⁡{1+aRW(a,Y):a>0}.It is known that GGLD⇒WNS⇒wfpp and the Opial property implies wfpp. Also
(2.9)R(Y)<2⟹MW(Y)>1⟹wfpp,(2.10)J(Y)<2⟹MW(Y)>1⟹wfpp.
First we will show that the Opial property is inherited from(X,|∥·∥|) to (X,∥·∥) and that (X,∥·∥) has GGLD if and only if (X,|∥·∥|) has GGLD. In order to achieve this, we need the following result shown in [7].Lemma 2.4.
Let(X,∥·∥) be a Banach space with a premonotone basis {en}. Then(1)
if{xn} converges weakly to x, limn∥xn-x∥=a if and only if limn|∥xn-x∥|=a,(2)
if{xn} converges weakly to 0 and limnlimr∥xn-xr∥=a, there exists a subsequence {yn} of {xn} such that limnlimr|∥yn-yr∥|=a.Lemma 2.5.
Let(X,∥·∥) be a Banach space with a premonotone basis {en}. If (X,|∥·∥|) has the Opial property, then (X,∥·∥) also has the Opial property, but the converse is false.Proof.
Let{xn} be weakly null in X and x∈X,x≠0. Then, by Lemma 2.4 and by (2.2),
(2.11)lim⁡sup⁡n‖xn‖=lim⁡sup⁡n|‖xn‖|<lim⁡sup⁡n|‖xn-x‖|≤lim⁡sup⁡n‖xn-x‖.
It is known that, for1<p<∞, (lp,∥·∥) has the Opial property. Consider any Γ-norm |∥·∥| in lp with the canonical basis {en}, and let δ>0 be such thatδ<((γ2p-γ1p)/γ1p)1/p. Then, for n≥2,
(2.12)|‖δe1+en‖|=max⁡[γ1(δp+1)1/p,γn]=γn,lim⁡n|‖en‖|=lim⁡nγn=lim⁡n|‖δe1+en‖|=1.
Thus, (lp,|∥·∥|) does not have the Opial property.Lemma 2.6.
Let(X,∥·∥) be a Banach space with a premonotone basis {en}. Then, (X,∥·∥) has GGLD if and only if (X,|∥·∥|) has GGLD.Proof.
Let{yn} be a weakly null normalized block basic sequence. By Lemma 2.4, limn∥yn∥ exists if and only if limn|∥yn∥| exists and in this case limn∥yn∥=limn|∥yn∥|. Also
(2.13)lim⁡nsup⁡i,j≥n‖yi-yj‖=lim⁡nsup⁡i,j≥n|‖yi-yj‖|.
The above equality follows immediately from the following inequality, for i, j≥n:(2.14)|‖yi-yj‖|≤‖yi-yj‖≤1γn|‖yi-yj‖|.
This proves the lemma.Now we will show that there exists a space with WNS such that with theΓ-norm it does not have WNS.Lemma 2.7.
LetX be the space c0 with the norm ∥x∥=sup|bi|+∑i=1∞ɛi|bi|, where x=∑i=1∞biei, ɛi>0 and ∑i=1∞ɛi<∞.Then X has WNS.Proof.
Let{xn}⊂X be a weakly null nonzero sequence. We may assume that x1≠0 and that there exists a block basic sequence {un}⊂X with ∥un-xn∥→n→∞0. Suppose that un=∑i=pnqnaiei with pn≤qn<pn+1 for n∈ℕ and that x1=∑i=1∞biei. Let k be such that ∑i=1kɛi|bi|=δ≠0. Let ε<δ/2, s>k with ∥Qsx1∥<ε and n>s. Then,
(2.15)‖x1-un‖+ε≥‖Psx1-un‖=max⁡(‖Psx1‖∞,‖un‖∞)+∑i=1sεi|bi|+∑i=pnqnεi|ai|≥‖un‖+δ.
Hence, limsupn∥x1-xn∥≥limsupn∥xn∥+δ/2 and {xn} cannot be a diametral sequence, since for a diametral sequence {xn} it is true that limn∥x-xn∥=diam{xn} for every x∈conv¯{xn}.Lemma 2.8.
LetΓ={γn}⊂(0,1) be an increasing sequence with limnγn=1. Then there is a space (X,∥·∥) with WNS, such that X with the Γ-norm  |∥·∥| does not have WNS.Proof.
Let{γnj}j be a subsequence of {γn} such that, if εnj=(1/3)(1/γnj-1), then ∑i=1∞εni<∞. Observe that εnj<εnj+1. Let X be the space c0 with the norm ∥x∥=sup|ai|+∑i=1∞εni|ani|+∑i=1,i≠nj∞(|ai|/2i), where x=∑i=1∞aiei. By Lemma 2.7, since ∑i=1∞(εni+1/2i)<∞, we know that X has WNS. Now let |∥·∥| be the Γ-norm in X with respect to {γi}. Let uj=enj; then γnj=1/(1+3εnj), and thus
(2.16)|‖uj‖|=γnj(1+εnj)<1,
and, if j<m,
(2.17)|‖uj-um‖|=max⁡(γnj(1+εnj+εnm),γnm(1+εnm))≤max⁡(γnj(1+2εnj),γnm(1+εnm))<1.
Therefore, since 0∈conv¯{un} and limn|∥un∥|=1, we have that diam|∥·∥|{un}=1. Also, if 0≤λi, ∑i=1n-1λi=1,
(2.18)|‖un‖|≤|‖∑i=1n-1λiui-un‖|≤1.
Hence, {un} is diametral in (X,|∥·∥|).The above example is another proof of the fact that for everyε>0 there are Banach spaces X and Y with d(X,Y)<1+ε so that X has WNS but Y does not.With regard to the coefficientMW(X), we will see that, if X has a premonotone basis and MW(X,∥·∥)>1, then MW(X,|∥·∥|)>1 and we will show a sufficient condition for the reverse implication. For this we need the following lemma.Lemma 2.9.
LetX be a Banach space with a basis. If
(2.19)RW1(a,X)=sup⁡{min⁡(liminf⁡‖un+y‖,liminf⁡‖un-y‖):un⟶w0,{un}⊂BXisablockbasicsequence,‖y‖≤aandsupportofyisfinitemin⁡(liminf⁡‖un+y‖,liminf⁡‖un-y‖):un⟶w0},
then RW(a,X)=RW1(a,X).Proof.
It is clear thatRW1(a,X)≤RW(a,X).
Now letε>0, x∈X, ∥x∥≤a, and {xn}⊂BX, with xn→w0such that min(limn∥xn+x∥,limn∥xn-x∥)>RW(a,X)-ε. By passing to a subsequence, we may assume that there exist a block basic sequence {un}⊂BX with ∥xn-un∥<ɛ and m∈ℕ such that ∥x-Pmx∥<ɛ and ∥Pmx∥<(1+ɛ)∥x∥≤a(1+ε). Then,
(2.20)‖Pmx1+ε+un‖≥‖xn+x‖-‖xn-un‖-‖x-Pmx‖-‖Pmx‖ε1+ε,‖Pmx1+ε-un‖≥‖xn-x‖-‖xn-un‖-‖x-Pmx‖-‖Pmx‖ε1+ε.
Let y=Pmx/(1+ε); then ∥y∥≤a, and we conclude that RW1(a,X)≥RW(a,X)-(3+a)ε, thus proving the assertion.Similarly one can prove that, ifX is a space with a basis, (2.21)R(Y)=sup⁡{lim⁡inf⁡n‖un+x‖:{x},{un}⊂BY,un⟶w0,{un}isablockbasicsequenceandsupportofxisfinite(lim⁡inf⁡n‖un+x‖:{x},{un}⊂BY,un⟶w0)}.It is known (see [6, 8]) that, if X, Y are Banach spaces, then(2.22)MW(X)≤MW(Y)d(X,Y),J(X)≤J(Y)d(X,Y).
So, if X is a Banach space with a basis, Γ={γn} with 0≤γn≤γn+1≤1 and γ1>1/MW(X), and Y is X with the Γ-norm, then MW(Y)≥MW(X)/d(X,Y)≥γ1MW(X)>1, and if γ1>1/MW(Y), MW(X)>1. Similarly, if γ1>J(X)/2, then J(Y)<2 and, if γ1>J(Y)/2, then J(X)<2. But the next proposition shows that in fact MW(X)>1 always implies MW(Y)>1. For the coefficient J, in general neither J(X)<2 implies J(Y)<2 nor the other way round, as we will see in Examples 2.16 and 2.17.Proposition 2.10.
Suppose thatX is a Banach space with a premonotone basis {en}. Then MW=MW(X,∥·∥)>1 implies that MW1=MW(X,|∥·∥|)>1.Proof.
LetRW(a,(X,∥·∥))=R(a) and RW(a,(X,|∥·∥|))=R1(a). Suppose that MW1=1. Then, R1(a)=1+a for every a>0. Let a>0 and 0<ε<a,y∈X with finite support, |∥y∥|≤a, and let {un} be a weakly null block basic sequence with |∥un∥|≤1 such that limn|∥un+y∥|>1+a-ε and limn|∥un-y∥|>1+a-ε. Then we may suppose that for every n,|∥un+y∥|>1+a-ɛ and |∥un-y∥|>1+a-ε. Hence,
(2.23)|‖un‖|≥1-ε,|‖y‖|≥a-ε.
We may also assume that the supports of y and un are disjoint. Let un=∑i=lnrnaiei and y=∑i=1rbiei.
Suppose that|∥un+y∥|=γmn∥∑i=mnrn(ai+bi)ei∥ for some mn≤rn. It is not possible that mn>r, because this would mean that
(2.24)1+a-ε≤|‖un+y‖|=γmn‖∑mnrnaiei‖≤|‖un‖|≤1.
So, by passing to a subsequence if necessary, we may assume that for every n we have that mn=i0≤r. Thus,
(2.25)1+a-ε≤|‖un+y‖|=γi0‖∑i=i0rn(ai+bi)ei‖≤γi0(‖∑i=lnrnaiei‖+‖∑i=i0rbiei‖)≤γi0γln+a.
Since limnγln=1, by passing to the limit, we obtain that
(2.26)γi0≥1-ε.
Similarly, there exists γi1≥1-ε with
(2.27)1+a-ε≤|‖un-y‖|≤γi1γln+a.
Suppose that i1≥i0, and let y0=∑i=i0rbiei and y1=∑i=i1rbiei. Then, since the basis is premonotone, 1+a-ε≤|∥un+y∥|=|∥un+y0∥|≤∥un+y0∥ and 1+a-ε≤|∥un-y∥|=|∥un-y1∥|≤∥un-y1∥≤∥un-y0∥; thus,
(2.28)a-ε≤|‖y0‖|≤|‖y‖|≤a.
Therefore,
(2.29)a-ε≤|‖y0‖|≤‖y0‖≤1γi0|‖y0‖|≤aγi0≤a1-ε
and |∥y0∥-a|≤max{ɛ,aɛ/(1-ɛ)}. Further, since γi0≤γln,
(2.30)1-ε≤|‖un‖|≤‖un‖≤1γln|‖un‖|≤1γln≤11-ε
and |∥un∥-1|≤ε/(1-ε).
Hence,(2.31)‖un‖un‖+y0‖y0‖a‖≥‖un+y0‖-|‖un‖-1|-|‖y0‖-a|≥1+a-ε-ε1-ε-max⁡{ε,aε1-ε}.
Similarly,
(2.32)‖un‖un‖-y0‖y0‖a‖≥1+a-ε-ε1-ε-max⁡{ε,aε1-ε}.
We deduce that min(liminfn∥(un/∥un∥)+(y0/∥y0∥)a∥,liminfn∥(un/∥un∥)-(y0/∥y0∥)a∥)≥1+a-ɛ-ε/(1-ε)-max{ε,aε/(1-ε)}, and letting ɛ tend to zero we obtain R(a)=1+a and MW=1.Examples2.14 and 2.17 exhibit spaces in which MW(X,∥·∥)=1 and MW(X,|∥·∥|)>1. There is however a special case in which MW(X,∥·∥)≥MW(X,|∥·∥|).Recall that a basis{en} of a Banach space X is 1-spreading if, whenever x=∑i=1∞aiei∈X and {eni}i is a subsequence of {en},∑i=1∞aieni∈X and ∥∑i=1∞aiei∥=∥∑i=1∞aieni∥.Proposition 2.11.
IfX is a Banach space with a premonotone 1-spreading basis, then MW(X,∥·∥)≥MW(X,|∥·∥|).Proof.
LetTm:X→X be the translation given by Tm∑i=1∞aiei=∑i=1∞aiei+m, and let a>0,y∈X, ∥y∥≤a, {un}⊂BX, where y has finite support and {un} is a weakly null block basic sequence such that min(∥y+un∥,∥y-un∥)>RW(a,(X,∥·∥))-ɛ for every n. Let m∈ℕ; then there exists N∈ℕ such that for n>N the supports of Tmy and un are disjoint, and thus, since the basis is 1-spreading, ∥y-un∥=∥Tmy-un∥ and ∥y+un∥=∥Tmy+un∥. Then, |∥Tmy∥|≤∥Tmy∥≤a, |∥un∥|≤∥un∥≤1 and for n>N(2.33)|‖Tmy-un‖|≥γm‖Tmy-un‖=γm‖y-un‖,|‖Tmy+un‖|≥γm‖Tmy+un‖=γm‖y+un‖.
Hence, RW(a,(X,|∥·∥|))≥γmRW(a,(X,∥·∥)), and, by passing to the limit as m tends to infinity, we get RW(a,(X,|∥·∥|))≥RW(a,(X,∥·∥)) and thus the desired result.Similarly to Propositions2.10 and 2.11 we can prove the following.Proposition 2.12.
Suppose thatX is a Banach space with a premonotone basis {en}. Then R=R(X,∥·∥)<2 implies that R1=R(X,|∥·∥|)<2 and, if the basis is premonotone and 1-spreading, then R=R(X,∥·∥)≤R(X,|∥·∥|)=R1.Corollary 2.13.
IfX is a Banach space with a premonotone 1-spreading basis, then MW1>1 if and only if MW>1; also R1<2 if and only if R<2.Next we will show an example of a space without a 1-spreading basis, such thatR=2, MW=1 but R1<2 and thus MW1>1.Example 2.14.
LetX be c0 with the following norm:
(2.34)‖{an}‖=|a1|+max⁡i≥2|ai|.
Let {en} denote the canonical basis of c0. Then for every a>0, ∥ae1+en∥=∥ae1-en∥=1+a, and thus R=2 and MW=1. On the other hand let x∈X with finite support and |∥x∥|≤1 and suppose that {un} is a block basic sequence with |∥un∥|≤1 for n∈N, un=∑i=mnrnaiei with mn≤rn<mn+1 and the support of x does not intersect the support of un. Then, for every m≥2 and n≥m,γm∥Qmx+un∥=max(γm∥Qmx∥c0,γm∥un∥c0)≤1,
(2.35)γ1‖x+un‖≤1+γ1‖un‖c0≤1+γ1γmn.
Thus, liminf|∥x+un∥|≤1+γ1. Hence R1≤1+γ1<2 and, by (2.9), MW1>1.With regard to the coefficientJ we have the following which is proved similarly to Proposition 2.11.Proposition 2.15.
Suppose thatX is a Banach space with a premonotone 1-spreading basis {en}.Then J(X,∥·∥)≤J(X,|∥·∥|).In general neitherJ(X,∥·∥)<2 implies J(X,|∥·∥|)<2 nor the other way round, as the following examples show.Example 2.16.
Let1>μ≥1/2 and X=ℝμ2⨁2l2, where ℝμ2=(ℝ2,∥·∥(μ)) and ∥(x1,x2)∥(μ)=max(|x1|,|x2|,μ(|x1|+|x2|)). Then, if μ≤γ2/(γ1+γ2), J(X)<2 but for every Γ, J(X,|∥·∥|)=2.
Sinceμ≥1/2, it is easy to see that for x=(x1,x2)∈ℝ2,
(2.36)μ1-2μ+2μ2‖x‖2≤‖x‖(μ)≤μ2‖x‖2.
Thus, d(l2,X)≤d(ℝ2,ℝμ2)=21-2μ+2μ2<2, and by (2.22), since J(l2)=2, we obtain thatJ(X)<2.
Now letΓ={γn}, μ≤γ2/(γ1+γ2), x=(1/γ1,1/γ2,0,0,…), and y=(-1/γ1,1/γ2,0,0,…). Then |∥x∥|=|∥y∥|=1 but |∥x+y∥|=|∥x-y∥|=2.This last example is another proof of the known fact that for everyɛ>0 there are Banach spaces X and Y with d(X,Y)<1+ε, J(X)<2 but J(Y)=2. In the following example we exhibit a space X with J(X)=2 such that J(X,|∥·∥|)<2.Example 2.17.
LetX=(l2,∥·∥), where, for x=(an)∈l2, ∥x∥=|a1|+(∑i=2∞ai2)1/2. Then J(X)=2,MW(X)=1, and, if Γ is such that γ2>1/2 and γ1/γ2<1/2, J(X,|∥·∥|)<2 and thus MW(X,|∥·∥|)>1.
Obviously if{en} is the canonical basis of l2, and a>0, then ∥ae1+en∥=∥ae1-en∥=1+a for n>1 and thus J(X)=2 and MW(X)=1.
Suppose now thatJ(Y)=J(X,|∥·∥|)=2.Then there exist sequences {xn},{yn}⊂BY such that limn|∥xn+yn∥|=limn|∥xn-yn∥|=2and {mn},{ln}⊂ℕ so that
(2.37)|‖xn+yn‖|=γmn‖Qmn(xn+yn)‖,|‖xn-yn‖|=γln‖Qln(xn-yn)‖.
By passing to a subsequence, we may assume that, for everyn∈ℕ, e1*(xn)≥0, e1*(yn)≥0 and e1*(xn)≥e1*(yn), and that the subsequence satisfies one of the next three cases:(1)
mn>1 and ln>1 for every n,(2)
mn=ln=1 for every n,(3)
mn=1 and ln>1 for every n.
Observe that(2.38)γm‖Qm(x+y)‖≥2-εimpliesγm‖Qm(x)‖≥1-εandγm‖Qm(y)‖≥1-ε.
Case 1.
LetZ=(l2,|∥·∥|Γ1), where |∥·∥|Γ1 is the Γ1={γi}i=2∞-norm. Then
(2.39)|‖Q2xn‖|Γ1≤1,|‖Q2yn‖|Γ1≤1,|‖Q2(xn+yn)‖|Γ1≥2-ε,|‖Q2(xn-yn)‖|Γ1≥2-ε.
Since 1/γ2<2, then, by (2.2), d(Z,l2)<2 and, by (2.22), since J(l2)=2,J(Z)<2 and this is a contradiction.Case 2.
Letɛ>0. Suppose that x=∑i=1∞aiei, y=∑i=1∞biei∈BY, a1>0,b1>0, a1>b1 and
(2.40)γ1(∑i=2∞(ai+bi)2)1/2≥2-ε-γ1(a1+b1),γ1(∑i=2∞(ai-bi)2)1/2≥2-ε-γ1(a1-b1).
Squaring both inequalities and adding them, since γ2(∑i=2∞ai2)1/2≤|∥x∥|, we get
(2.41)2γ12γ22≥γ12∑i=2∞(ai2+bi2)≥(2-ε-γ1a1)2+γ12b12≥(2-ε-γ1a1)2.
By passing to the limit as ɛ→0, since γ1a1≤1, we obtain that 2(γ1/γ2)≥2-γ1a1≥1, and this contradicts γ1/γ2<1/2.Case 3.
Let2>ε>0. Suppose that x=∑i=1∞aiei,y=∑i=1∞biei∈BY,a1>0,b1≥0,a1>b1 and |∥x+y∥|=γ1(|a1+b1|+(∑i=2∞(ai+bi)2)1/2)≥2-ε and |∥x-y∥|=γm(∑i=m∞(ai-bi)2)1/2≥2-ɛ.Then, by (2.38),
(2.42)γm(∑i=m∞ai2)1/2≥1-ε.
Since inl2 if u,v∈Bl2, one has that ∥u-v∥≥δ implies ∥u+v∥≤21-(δ/2)2, then γm(∑i=m∞(ai+bi)2)1/2≤4ɛ-ɛ2.Also
(2.43)2-ε≤γ1(|a1+b1|+(∑i=2m-1(ai+bi)2)1/2+(∑i=m∞(ai+bi)2)1/2)≤γ1|a1+b1|+γ1(∑i=2m-1(ai+bi)2)1/2+γ1γm4ε-ε2≤γ1|a1+b1|+γ1(∑i=2m-1(ai+bi)2)1/2+4ε-ε2.
Let ϕ=ε+4ε-ε2; then
(2.44)2-ϕ≤γ1|a1+b1|+γ1(∑i=2m-1(ai+bi)2)1/2.
By (2.38),γ1|a1|+γ1(∑i=2m-1ai2)1/2≥1-ϕ, and since, for A,B>0,(2.45)(A+B)1/2-A1/2=B(A+B)1/2+A1/2,
we have, using (2.42), that
(2.46)1≥γ1|a1|+γ1(∑i=2∞ai2)1/2≥γ1|a1|+γ1(∑i=2m-1ai2+(1-ε)2γm2)1/2=γ1|a1|+γ1(∑i=2m-1ai2)1/2+γ1(1-ε)2γm21(A+B)1/2+A1/2≥1-ϕ+γ1(1-ε)2γm21(A+B)1/2+A1/2,
where A=(∑i=2m-1ai2)1/2 and B=(1-ɛ)2/γm2.But
(2.47)(A+B)1/2+A1/2≤(1γ22+(1-ε)2γm2)1/2+1γ2≤3γ2;
therefore,
(2.48)γ1<γ1γm2≤ϕ(1-ε)23γ2,
and, taking the limit as ε→0, we get γ1=0 which is a contradiction.
Hence,J(X,|∥·∥|)<2, and, by (2.10), we have that MW(X,|∥·∥|)>1.



---

*Source: 287145-2011-05-19.xml*