DEPENDENT CHOICES AND WEAK COMPACTNESS

DEPENDENT CHOICES AND WEAK COMPACTNESS CHRISTIAN DELHOMM′E,MARIANNE MORILLON

ABSTRACT.We work in set-theory without the Axiom of Choice

ZF.We prove that the principle of Dependent Choices(DC)im-

plies that the closed unit ball of a uniformly convex Banach space

is weakly compact,and in particular,that the closed unit ball of a

Hilbert space is weakly compact.These statements are not provable

in ZF,and the latter statement does not imply DC.Furthermore,

DC does not imply that the closed unit ball of a re?exive space is

weakly compact.

Mathematics Subject Classi?cation:primary03E25,04A25;

secondary46,54.

Keywords:Axiom of Choice,principle of Dependent Choices,

weak topology,re?exive spaces.

1.Introduction

We work in set-theory without Axiom of Choice ZF,and we denote byωthe set of natural numbers.In this paper,normed spaces(as de?ned for example in[2],De?nition1.2p.63)are real normed spaces, and they are endowed with the norm metric.A metric space is said to be complete when every Cauchy?lter of this space converges(see Remarks6and7).A Banach space is a normed space which is complete. The continuous dual of a normed space(E, . )is the vector space E?of real linear functionals on E which are bounded on the closed unit

ball of E,and E?is endowed with the dual norm .

?:for every

f∈E?, f

?:=sup{f(x): x ≤1}.The second dual of E is the

normed space E??.For every x∈E,we denote by?x the evaluation at point x,i.e.the mapping E?→R such that for every f∈E?,?x(f)=f(x).The natural map j E:E→E??,given by j E(x)=?x,is

linear,and continuous since j E(x)

??≤ x .Using the Hahn-Banach

axiom,j E can be proved isometric,i.e.?x∈E j E(x)

??= x (see

[2]Corollary6.7,p.79),but this is not provable in ZF,since there are models of ZF with in?nite dimensional normed spaces E such that E??={0},see Remark5).The usual de?nition of“re?exivity”for a normed space E(see[2]p.89De?nition11.2)relies on the fact that j E is isometric,so we will reformulate this de?nition in ZF and we

1

2 C.DELHOMM ′E,

M.MORILLON will call it “simple re?exivity”.The weak topology of E is the coarsest topology on E for which every f ∈E ?is continuous :it is generated by the sets {x ∈E :f (x )<λ},λ∈R and f ∈E ?and it is denoted by σ(E,E ?)(see [2]De?nition 1.1p.124).The weak*topology of E ?(see [2]De?nition 1.1pp.124-125)is the coarsest topology on E ?such that for every x ∈E ,?x is continuous :it is generated by the sets {f ∈E ?:f (x )<λ},λ∈R and x ∈E ,and it is denoted by σ(E ?,E ).A topological space X is compact if every non-empty set of closed subsets of X with the ?nite intersection property has a non-empty intersection.In set-theory with the Axiom of Choice ZFC ,the re?exivity of E is known to be equivalent to the compactness of its closed unit ball for its weak topology (see [2]Theorem 4.2p.132),but this equivalence is not provable in ZF (see Remarks 2,4and 5),so we shall consider another notion of re?exivity,which we call “compact re?exivity”.

Let us state these two notions of re?exivity for a normed space E :(Simple)Re?exivity :The natural mapping j E from E to its second dual E ??is onto and isometric.

Compact Re?exivity :The closed unit ball of E is compact for the weak topology.

Note that the classical proof of the following statement of Re?exive Compactness relies on Alaoglu’s theorem (see [2]Theorem 3.1pp.130-131),which is equivalent (within ZF )to many other classical state-ments,e.g.the Boolean Prime Ideal Theorem (see Howard and Ru-bin [4]pp.21-27);this last statement is not provable in ZF (see Jech [5]),hence Alaoglu’s theorem is not provable in ZF either.

RC (Re?exive Compactness):The closed unit ball of a re?exive normed space is compact for the weak topology.

A (Alaoglu):The closed unit ball of the continuous dual of a normed space is compact for the weak*topology.

We now consider some geometric properties of normed spaces.A normed space (E, . )is a prehilbert space when there exists an inner product .,. :E ×E →R such that for every x ∈E , x = x,x .A Hilbert space is a complete prehilbert space.A normed space E is uniformly convex (see [1]p.189)if the modulus of uniform convexity of E ,δE :R ?+?→R +de?ned below,satis?es ε>0?δE (ε)>0.δE (ε):=inf 1? x +y 2 :x,y ∈ΓE and x ?y ≥ε

DEPENDENT CHOICES AND WEAK COMPACTNESS3 Every prehilbert space is uniformly convex(see[1]pp.189-190).We now consider Re?exive Compactness particularized to uniformly convex Banach spaces,and particularized further to Hilbert spaces:

RCuc(Re?exive compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is weakly com-pact.

RCh(Re?exive Compactness for Hilbert spaces):The closed unit ball of a Hilbert space is weakly compact.

https://www.wendangku.net/doc/7e11318815.html,ing projections on closed convex subsets in a Hilbert space(see Lemma3in[3]),one can prove in ZF that every Hilbert space is simply re?exive.Hence RC implies RCh;in particular RCh does not imply DC.

In Fossy and Morillon[3],it is proved that RCh implies the following

set-theoretic axiom AC?n

ωwhich is not provable in ZF(see[5]);in

particular,the statements RC and RCh are not provable in ZF either.

AC?n

ω(Countable Axiom of Choice for?nite sets):If(A n)n∈ωis a

sequence of non-empty?nite sets,then

n∈ω

A n=?.

Remark2.Thus,though in ZF every Hilbert space is simply re?exive, there are models of ZF in which some Hilbert spaces are not compact re?exive.Hence simple re?exivity does not imply compact re?exivity.

Now the following question is natural:

Question.Is there a principle of“Countable Choice”which implies the axiom RCh?

In this paper,we prove that the following principle of Dependent Choices implies RCuc(thus it implies RCh too):

DC(principle of Dependent Choices):If E is a non-empty set and R is a binary relation on E satisfying

?x∈E?y∈E xRy,

then there exists a sequence(x n)n∈ωsuch that for every n∈ω,x n Rx n+1.

We shall also observe that the principle of Dependent Choices does not imply RC(see Remark4).

Note that BPI does not imply DC,and that DC does not imply BPI (see[4],or[5]).

4 C.DELHOMM ′E,

M.MORILLON 2.The principle of Dependent Choices implies RCuc

Notation.Consider a (real)normed space (E, · ).For each non-negative real number r ,Γ(0,r )denotes the closed ball of center 0and radius r ,i.e.{z ∈E : z ≤r };the closed unit ball Γ(0,1)is denoted by ΓE .Given two real numbers r and r such that 0≤r ≤r ,the crown {z ∈E :r ≤ z ≤r }is denoted by D (0;r,r ).

Given a normed space E ,we denote by T E the set of ?nite unions of closed convex subsets of ΓE .Notice that (T E ,∩,∪)is a lattice of subsets of ΓE and that each closed set of ΓE for the weak topology is an intersection of elements of T E .A ?lter of T E is any non-empty set F of non-empty elements of T E such that the intersection of any two elements of F is in F and such that any element of T E which is a superset of an element of F is in F too.

For each set F of subsets of ΓE ,let R (F )=inf r ∈R :0≤r ≤1and ?F ∈F ,Γ(0,r )∩F =? .When F has the ?nite intersection property,F ∪{Γ(0,r ):R (F )

The following lemma is an easy consequence of the de?nitions :Lemma 1.Let E be a normed space and F be a ?lter of T E .

—For every real numbers r and r such that 0≤r

—For any ?lter F of T E extending F c :R (F )=R (F )and (F )c =F . Lemma 2.Given a uniformly convex normed space E with modulus of uniform convexity δE ,consider real numbers η>0and r,r such that 0

subset C of the crown D (0;r,r )is less than or equal to η.

Proof.Assume by contradiction that some convex subset C of D (0;r,r )contains two points x and y such that x ?y >η.Then,from the de?nition of δE ,it follows that x +y 2 < 1?δE (ηr ) r ;but x +y 2 ≥r ,since C is convex. Theorem.Given a uniformly convex Banach space E ,let F be a ?lter of T E .The principle of Dependent Choices implies that the set ∩F is non-empty.

DEPENDENT CHOICES AND WEAK COMPACTNESS 5

Proof.We prove the existence of a Cauchy ?lter G of T E (i.e.a ?lter containing sets of arbitrary small diameter)extending the circled ?lter F c associated to F (thus,since the elements of G are closed and E is complete,∩G =?and a fortiori ,∩F =?).Denote by δE the modulus of uniform convexity of E .Let R =R (F ).If R =0,then F c is already Cauchy.Now assuming that R >0,for each n ∈ω,

let r n and r n be positive real numbers such that r n 0such that 2αn n ≤δE (12(n +1)R )and let r n =R ?αn and r n =R +αn ).Let S denote the

set of ?nite mappings s ?ω×T E such that

i /for every n ∈domain(s ),s (n )?D (0;r n ,r n ),ii /F c ∪range(s )has the ?nite intersection property.

Every element of S admits a proper extension in S :given s ∈S and n /∈domain(s ),it follows from Lemma 1that some element F of the

?lter F s of T E generated by F c ∪range(s )is a subset of D (0;r n ,r n );

then,given closed convex subsets C 1,...,C m of E such that F =C 1∪...∪C m ,observe that for some i ∈{1,···,m },C i meets every element of F s ,so that s ∪{(n,C i )}is a proper extension of s in S .Now,invoking DC ,get an increasing sequence (w.r.t.proper extension)(s n )n ∈ωof T E ,and observe that,given s =∪{s n :n ∈ω},Lemma 2implies that F c ∪range(s )generates a Cauchy ?lter of T E . Corollary.DC =?RCuc .

Proof.Given a uniformly convex Banach space E ,let H be a non-empty set of weakly closed sets of ΓE with the ?nite intersection property.Since each such closed set is an intersection of elements of T E ,∩H =∩F ,where F ={F ∈T E :F ?H,for some H ∈H};but F is a ?lter of T E ,hence ∩F =?,according to the theorem above. Remark 3.RCuc does not imply A ,since RCuc follows from DC ,which does not imply BPI ,and hence,does not imply A either.

Remark 4.Pincus and Solovay [6]have built a model M of (ZF+DC)in which every ?nitely additive measure is trivial.This means that,given any set I ,for every ?nitely additive mapping m :P (I )→R ,there exists a family (λi )i ∈I of real numbers such that,for every subset A of I :m (A )= i ∈A λi .It follows that the continuous dual of ∞(I )is equal to 1(I ).Thus,in this model M ,every 1(I )is a re?exive normed space.

Besides,the closed unit ball Γof 1(I )is never weakly compact when I is in?nite :in fact,denoting by P f (I )the set of ?nite subsets of I ,

6 C.DELHOMM ′E,

M.MORILLON for each H ∈P f (I ),let

F H = f ∈Γ: k ∈I

f (k )=1and ?k ∈H,f (k )=0 .

Each F H is a weakly closed set of Γ,and when I is in?nite,the family {F H :H ∈P f (I )}has the ?nite intersection property,but {F H :H ∈P f (I )}is empty.

Summing up,in the model M ,the space 1(ω)is re?exive and sep-arable but its closed unit ball is not weakly compact.Hence “simple re?exivity”does not imply “compact re?exivity”,even in the case of separable spaces.Moreover,since the model M satis?es DC ,DC does not imply RC ,even for separable re?exive spaces.

Remark 5.“Compact re?exivity”does not imply “simple re?exivity”since,in every model of ZF where Hahn-Banach Theorem fails (for in-stance the model M above),there exists an in?nite dimensional normed space E such that E ?={0}:such an E is not re?exive although ΓE is weakly compact.

Let us now consider the following consequence of DC :

AC ω(Countable Axiom of Choice):If (A n )n ∈ωis a sequence of non-empty sets,then n ∈ωA n =?.

Remark 6.Say that a metric space (X,d )is sequentially complete when every Cauchy sequence converges.So every complete metric space is sequentially complete.In (ZF+AC ω),hence in (ZF+DC),every sequentially complete metric space is complete.

Remark 7.A set X is Dedekind-?nite when there exists no one-to-one mapping from ωto X .There are models of ZF (for example Cohen’s ?rst model,see [5])with a Dedekind-?nite dense subset A of R .The metric subspace A is sequentially complete but it is not complete.We know no answer to the following questions :

Question 1.Are the statements A and RC equivalent ?

Question 2.Does AC ωimply RCh or RCuc ?

Question 3.Are RCh and RCuc equivalent ?

DEPENDENT CHOICES AND WEAK COMPACTNESS 7

BPI A /|||||||||||||||||g g g g g g g g g g g g g g g g g g C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C DC /p p p p p p p p p p p p p p p p p p \HB /RC /.-,()*+?g g g g g g g g X X X X X X X X X X X X X X X X X X X X X X X X X AC ω//.-,()*+?m m m m m m /.-,()*+?????????????RCuc RCh m m m m m m m m m m m m m /.-,()*+?\AC ?n ω

\Note :Uniformly convex spaces are simply re?exive (work in progress).Thus RC implies RCuc ;in particular,like RCh ,RCuc fails to imply DC .(Cf.Abstract and Remark 1.)

References

[1]Beauzamy, B.,Introduction to Banach spaces and their geometry ,North-

Holland,Amsterdam,1985.

[2]Conway,J.B.,A course in functional analysis ,2nd ed.,Springer-Verlag,1990.

[3]Fossy,J.,and M.Morillon,“The Baire Category Property and some Notions

of Compactness”,J.London Math.Soc.(2)57(1998),pp.1-19.

[4]Howard,P.,and J.Rubin,Consequences of the Axiom of Choice ,Mathematical

Surveys and Monographs,Vol.59,AMS,1998.

[5]Jech,T.,The Axiom of Choice ,Studies in Logic and the Foundations of Math-

ematics,North-Holland,1973.

[6]Pincus,D.,and R.M.Solovay,“De?nability of measures and ultra?lters”,J.

Symbolic Logic 42(1977),pp.179-190.

ERMIT,Dpartement de Mathmatiques et Informatique,Universit de La Runion,15avenue Ren Cassin -BP 7151-97715Saint-Denis Messag.Cedex 9FRANCE E-mail address ,Christian Delhomm′e :delhomme@univ-reunion.fr

E-mail address ,Marianne Morillon:mar@univ-reunion.fr

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（3）滴滴专车的车型要求： 1、车辆颜色：黑白银金棕（商务）、黑（舒适）。 2、车龄：车龄小于3年（具体以当地网约车政策要求为准），无明显伤痕、无改装。 3、车辆保险：交强险、商业险（第三者责任险100万以上；司乘座位每座最低2万；均含不计免赔）。 4、符合所在城市准入车型。 《专车新政》的第十五条对符合第十四条规定条件的驾驶员，由驾驶员提出申请，按规定经考核合格的，由所在地设区的市级道路运输管理机构发放类别为预约出租汽车的《道路运输从业人员从业资格证》。 https://www.wendangku.net/doc/7e11318815.html,

合肥艺伟汽车服务有限公司是一家专注于网约车一站式服务的本土平台，公司一直秉承以客户为中心、以市场为导向，紧紧跟随时代创新的脚步，以标准化服务体系和可视化服务内容，赢得市场所有用户的高度评价，也生发出了所有司机伙伴的强烈使命感！艺伟的乘客是幸福的，艺伟的司机是热忱的，艺伟的服务是长情的！ 合肥艺伟汽车服务有限公司历经四载发展，已经成为合肥地区最具竞争力和感召力的网约车服务窗口，换来了滴滴公司对于艺伟更多的信任和委以重任！我们将持续打造精英司机队伍，为更多人实现创收和梦想！合肥地区网约车运行证已近饱和，欢迎各位想要围城筑爱、同心护航的司机朋友们，选择艺伟，一起同行！更多详情请点击官网合肥艺伟汽车服务有限公司进行进一步咨询了解。 https://www.wendangku.net/doc/7e11318815.html,

地层时代中英文对照

圭亚那盆地地层时代中英文对照 Cainozoic 新生界 Tertiary 第三纪 Pleistocene 第三纪更新世 Miocene 第三纪中新世 Lower Miocene 第三纪下中新世 Oligocene 第三纪渐新世 Eocene 第三纪始新世 Middle Eocene第三纪中始新世 Lower Eocene第三纪下始新世 Paleocene 第三纪古新世、古近系 Cretaceous 白垩纪 Senonian 森诺阶 Campanian 坎帕阶 Santonian 桑托阶 Coniacian 科尼亚克阶（晚白垩纪第三期）Turonian 白垩纪土仑阶（晚白垩纪第二期）Cenomanian 森诺曼阶 Albian 阿尔比阶 Aptian 阿普第阶 Barremian 巴列姆阶 Precambrian 前寒武纪地层

Cainozoic 新生界 Quaternary 第四系Quaternary period 第四纪Paleocene 第三纪古新世、古近系 Mesozoic中生界 Cretaceous 白垩系 Jurassic 侏罗系 Triassic 三叠系 LateTriassicepoch 晚三叠世 Paleozoic 古生界 Permian 二叠系 LatePermianepoch Early Permian 早二叠世 Carboniferous 石炭系石炭纪 Late Carboniferous Devonian 泥盆系泥盆纪upperDevonionseries 上泥盆统 siluric 志留系Silurian 志留纪 Ordovician 奥陶系奥陶纪 Cambrian 寒武系寒武纪 Proterozoic era 元古代

科学翻译

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大数据时代英文翻译

Era of Big Data is a woman's age; women in the gene can accumulate and deal with big data/ women are born to accumulate and deal with big data. Many men and children, in fact, have been wondering about this special ability of women. Like, as a child, just as soon as you entered the house your Mother said immediately in a suspicious tone: “Liu zhijun, you didn’t do well in the exam today, did you.” Another example, you just have a glance at the mobile phone, your wife laughs: “Does Er gou the next door ask you to play games?” One more ex ample, when you close the door and make a phone call, your girlfriend will cry: “Who are shot in bed?” They are sometimes right, sometimes wrong. However, On the whole, the accuracy rate is higher than chance level. When they are wrong, men would sneer women always give way to foolish fancies; when they are right, men would say women are sensitive animal maybe with more acute sensory organs. Anyway, that is a guess. It has already scared man that overall accuracy rate is higher than the random level. In order to adapt to this point, the male also developed a very strong skills against reconnaissance. This part is beyond the scope of this article, so no more details about it. Some studies, such as Hanna Holmes’s paper, have indi cated that the white matter of the female’s brain is higher than that of the male. So they have very strong imagination of connecting things together. Some recent studies have shown that women are better than men in the "date" memory. That is the reason why they are able to remember all the birthdays, anniversaries, and even some of the great day of unimportant friends. No matter whether these results are true or not, I am afraid that this is not women's most outstanding ability. Women's most remarkable ability is a long-term tracking of some seemingly unimportant data to form their own baseline and pattern. Once the patterns of these data points are significantly different from the baseline she is familiar with, she knows something unusual. In their daily life, women do not consider the difference between causality and correlation. They believe in the principle: "There must be something wrong out of something unusual." People who talk about big data often take Lin Biao as an example. Lin Biao recorded some detailed and unimportant data after a battle. Such as seized guns, the proportion of rifles and pistols, the age levels of war prisoners, seized grain, whether they are sorghum or millet, etc., all of which were unavoidably recorded in the book. Others laughed at him. But later, he determined where the enemy headquarters were according to these data. What women do is almost the same. A girl A has a secret crush on boy B, but she usually doesn’t contact him directly. Two days later, I asked her if she wanted to ask him to have dinner together. She said he was playing. I wondered “how do you know that?” She said that boy B usually is on the line Gmail at 8:00 am, away status at8:30am, for he goes out to buy coffee and breakfast, on line again at 9:00am, busy status, for he is at work, away again at12:30am for lunch, on line for whole evenings, maybe for reading or playing games. His buddy C is on line at10:00 am, still online till 2:00am next day. He is a boy who gets up late and stays up late. His buddy D is on line for the most of the day. However, the most important pattern is that there are 2-3 days per week, during which they would be offline or away for 3-4 hours together. Conclusion: they are playing together.

滴滴快车运营模式探析

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