The Schur l1 Theorem for filters

THE SCHUR ℓ 1 THEOREM F OR FIL TERS ANTONIO A VILES LOPE Z, BERNARDO CASCALES SALINAS, VLADIMIR KADETS AND ALEXANDER LEONOV Abstract. W e study classes of filters F on N such that weak and strong F -con vergence of seq uences in ℓ 1 coincide. W e study also analogue of ℓ 1 weak sequential completeness theo r em for filter conv ergenc e . 1. Preliminaries Ev ery theorem of C lassical Analysis, F unc tiona l Analysis or of the Measure Theory that states a prop erty of sequen ces leads to a class of filters for whic h this theorem is v alid. Sometimes suc h class of filters is trivial (say , all filters or the filters with coun table base), but in sev eral cases this appro ac h leads to a new class of filters, and the c haracteriza- tion of this class can b e a v ery non-trivial task. Among suc h no n-trivial examples there are Leb esgue filters (f or whic h the Leb esgue dominated con v ergence theorem is v alid), Egorov filters which corresp ond to the Egoro v theorem on almost uniform con vergenc e [7], and those filters F for whic h eve ry w eakly F conv ergen t seque nce has a norm-b o unded subseque nce [6 ]. One of the r easons t o study such questions is that they bring a new ligh t to the classical results. Sa y , it is kno wn, that the dominated con v ergence theorem can b e deduced fr om the Egoro v theorem. The question, whether the con v erse is true has no sense in the classic al con text: if b oth the theorems are true, ho w one can see that one of them is not deducible from the other one? But if one lo oks at the corresp onden t classes of filters, the pro blem make s sense a nd in fa ct there a r e Leb esgue filters whic h are not Egor ov ones ( in particular the statistical con ve rgence filter). In this pap er w e study the Sch ur theorem on coincidence of w eak and strong con v ergence in ℓ 1 in a general setting when the ordinary con v ergence of seque nces is substituted by a filter con ve rgence. W e sho w that for some filters this theorem is v alid and for some is no t and giv e necessary conditions and sufficien t conditions (close one to another) for its v alidit y . After that w e consider the Sc hur theorem Date : January 12, 2007 . The work of the third a uthor was supp orted by the Seneca F oundation, Murcia. Grant no. 0212 2/IV2/05 . 1 2 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV for ultrafilters. W e also study a r elated problem o f w eak sequen tial completeness for filter conv ergence. Recall that a filter F on a set N is a not- empty collection of subsets of N satisfying the follow ing axioms: ∅ / ∈ F ; if A, B ∈ F then A T B ∈ F ; and for ev ery A ∈ F if B ⊃ A then B ∈ F . All o v er the pap er if the con trary is not stated directly we consider filters on a coun table set N . Sometimes for simplicit y w e put N = N . A sequence ( x n ) , n ∈ N in a to p ological space X is said to b e F - c onver gent to x (and we write x = F - lim x n or x n → F x ) if for ev ery neighborho o d U of x the set { n ∈ N : x n ∈ U } b elongs to F . In particular if one take s as F the filter of those sets whose comple- men t is finite (the F r ´ echet filter ), then F -conv ergence coincides with the ordinary one. The nat ural ordering on the set of filters on N is defined as follow s: F 1 ≻ F 2 if F 1 ⊃ F 2 . If G is a cen tered collection of subsets (i.e. all finite in tersections of the elemen ts of G are non-empt y), then there is a filter containing all the elemen ts of G . The smallest filter, con taining all the elemen ts of G is called the filter gen er ate d b y G . Let F b e a filter. A collection of subsets G ⊂ F is called the b ase of F if for ev ery A ∈ F there is a B ∈ G such that B ⊂ A . A filter F on N is said to b e fr e e if it dominates the F r ´ ec het fil- ter. All the filters b elo w are supp osed to b e free. In particular ev ery ordinary conv ergen t sequence will b e automatically F -conv ergent. A maximal in t he natural ordering filter is called an ultr afilter . The Zorn lemma implies that ev ery filter is dominated b y an ultrafilter. A filter F on N is an ultrafilter if a nd only if for ev ery A ⊂ N either A or N \ A b elongs to F . A subset of N is called stationary with resp ect to a filter F (or just F - stationary) if it has nonempt y inters ection with each mem b er of the filter. D enote the collection o f a ll F -statio na r y sets b y F ∗ . F or an I ∈ F ∗ w e call the collection of sets { A ∩ I : A ∈ F } the tr ac e of F on I (whic h is eviden tly a filter on I ), a nd by F ( I ) w e denote the filter on N generated b y the trace of F on I . Clearly F ( I ) dominates F . An y subset o f N is either a mem b er of F or the complemen t o f a mem b er of F o r the set a nd its complemen t are b oth F -stationary sets. F ∗ is precisely the union of all ultrafilters do minat ing F . F ∗ is a filter base if and only if it is equal to F and F is an ultrafilter. Theorem 1.1. L et X b e top olo gic al sp ac e, x n , x ∈ X and let F b e a filter on N . Then the fol lowing c onditions ar e e q uiva lent (1) ( x n ) is F -c on v er gent to x ; (2) ( x n ) is F ( I ) -c onv e r gent to x for ev e ry I ∈ F ∗ ; (3) x is a cluster p oint of ( x n ) n ∈ I for e very I ∈ F ∗ . Pr o of. Implications (1) ⇒ (2) and (2) ⇒ (3) a re eviden t. Let us pro v e that (3) ⇒ (1 ). Supp o se x n do not F -conv erge to x . Then there is THE SCHU R ℓ 1 THEOREM FOR FIL TERS 3 suc h a neigh b orho o d U of x that in each A ∈ F there is a j ∈ A suc h t hat x j 6∈ U . Consequen t ly I = { j ∈ N : x j 6∈ U } is statio nary and x is not a cluster p oin t of ( x n ) n ∈ I .  More ab out filters, ultra filters and their applications one can find in most of adv a nced General T op o logy textb o oks, fo r example in [10]. F or the standard Ba nac h space terminology w e refer to [8]. All the spaces, functionals and op erators (althoug h this do es not matter) are assumed t o b e ov er the field of reals. Before w e pass to the main results let us recall some notations and geometric prop erties of ℓ 1 . Denote b y e n the n - th elemen t of the canonical basis of ℓ 1 and b y e ∗ n the n -th co ordinate functional on ℓ 1 . In this notations for ev ery x ∈ ℓ 1 w e ha v e x = X n ∈ N e ∗ n ( x ) e n . Recall that e n are separated f r o m 0 b y the functional f ( x ) = P n ∈ N e ∗ n ( x ) , i.e. 0 is not a w eak cluster p oin t of ( e n ) . The fo llowing lemma can b e easily extracted from t he blo c k-basis selection metho d (see [8], v olume 1). W e giv e t he pro of for completeness. Lemma 1.2. L et y n ∈ ℓ 1 , inf n ∈ N k y n k = ε 0 > ε > 0 and let { m ( n ) } b e an incr e as i n g se quenc e of na tur als. Denote z i = P k ∈ ( m ( i ) ,m ( i +1)] e ∗ k ( y i ) e k . I f under these n otations sup n ∈ N k y n − z n k < ε/ 2 (i.e. ( y n ) is a s m al l p erturb ation of the blo c k-b asis ( z n ) ) then ( y n ) is e quivalent to the se quenc e ( k y n k e n ) and c onse q uen tly 0 is n o t a we a k cluster p oint of ( y n ) . Pr o of. W e m ust find c 1 , c 2 > 0 suc h tha t for ev ery collection of scalars a n c 1 X n ∈ N | a n |k y n k ≤      X n ∈ N a n y n      ≤ c 2 X n ∈ N | a n |k y n k . The upp er estimate with c 2 = 1 follo ws immediately f r o m the trian- gular inequalit y . The low er one holds with c 1 = 1 − ε 0 /ε      X n ∈ N a n y n      ≥ X n ∈ N | a n |k z n k − X n ∈ N | a n |k y n − z n k ≥ X n ∈ N | a n |k y n k − 2 X n ∈ N | a n |k y n − z n k ≥  1 − ε ε 0  X n ∈ N | a n |k y n k .  2. Simplified Schur p ro p er ty for fil ters There a re sev eral natural w ay s to g eneralize t he Sc hur theorem for filters instead of sequences . Let us start with the one leading t o a class of filters which w e are able to c har a cterize completely in com binatorial terms. 4 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV Definition 2.1. A filter F on N is said to b e a simple Schur filter (or is said to hav e the simp l i fi e d Schur pr op erty ) if for ev ery co ordinate- wise conv ergent to 0 sequence ( x n ) ⊂ ℓ 1 if ( x n ) w eakly F -conv erges to 0, then F - lim k x n k = 0 . F or an infinite set I ⊂ N let us call a b l o cking of I a disjoin t partition D = { D k } k ∈ N of I in to non-empty finite subsets. Definition 2.2. A filter F on N is said to b e blo c k -r esp e cting if for ev ery I ∈ F ∗ and for eve ry blo c king D of I there is a J ∈ F ∗ , J ⊂ I suc h that | J ∩ D k | = 1 fo r all k , where the “mo dulus” o f a set stands for the n umber of elemen ts in the set. R emark 2.3 . If in the definition ab ov e one writes (2.1) ∀ k ∈ N | J ∩ D k | ≤ 1 instead of | J ∩ D k | = 1 , one will obtain an equiv alen t definition. R emark 2.4 . If F is blo ck - r especting, then F ( J ) for ev ery J ∈ F ∗ is also blo ck - resp ecting. Lemma 2.5. L et F b e a blo c k - r esp e cting filter a nd let ( x n ) ⊂ ℓ 1 form a c o or dinate-wise c onver gent to 0 s e quenc e, which do es not F - c onve r ge to 0 in n o rm. Then ther e is a J ∈ F ∗ , such that the se quenc e ( x n ) , n ∈ J is e quivalent to ( a i e i ) , wher e e i form the c anonic al b asis of ℓ 1 , a i ≥ 1 . Pr o of. D ue to the Theorem 1.1 there is a n I ∈ F ∗ suc h that inf n ∈ I k x n k > ε > 0 . Fix a decreasing sequence of δ k > 0 , P k ∈ N δ k ≤ ε/ 8 . Using the definition of ℓ 1 let us select an increasing sequence of naturals ( m ( n )) and suc h that for eve ry n ∈ N (2.2) X k ≥ m ( n ) | e ∗ k ( x n ) | < δ n and using the co ordinate-wise con v ergence of x n to 0 select an increas- ing sequence of integers ( n i ) suc h that n 0 = 0 , D i := ( n i − 1 , n i ] ∩ I 6 = ∅ and for ev ery i ∈ N and j ≥ n i +1 (2.3) X k ≤ m ( n i ) | e ∗ k ( x j ) | < δ i . T aking in a ccount the respect whic h F has to the blo c ks D i let us select a J = { j 1 , j 2 , . . . } ∈ F ∗ , J ⊂ I suc h that j i ∈ ( n i − 1 , n i ] for all i ∈ N . Since J ∈ F ∗ , either J 1 = { j 1 , j 3 , j 5 . . . } or J 2 = { j 2 , j 4 , j 6 . . . } is an F -stationary set as w ell. Let, say , J 2 ∈ F ∗ . Let us sho w that in fact v ectors y i = x j 2 i are small p erturbat io ns of the blo c k- basis z i = P k ∈ ( m ( n 2( i − 1) ) ,m ( n 2 i )] e ∗ k ( y i ) e k , whic h due to the Lemma 1.2 completes the pro o f . So: THE SCHU R ℓ 1 THEOREM FOR FIL TERS 5 k y i − z i k = X k ≤ m ( n 2 i − 2 ) | e ∗ k ( x j 2 i ) | + X k >m ( n 2 i ) | e ∗ k ( x j 2 i ) | . T aking in to accoun t inequalities (2 .2), (2 .3) and that j 2 i ∈ ( n 2 i − 1 , n 2 i ] , w e g et k y i − z i k ≤ 2 δ j 2 i whic h implies the condition of Lemma 1.2.  Theorem 2.6. A filter F on N has the si m plifie d Sc hur pr op erty if and only if F is blo ck-r esp e cting. Pr o of. The “if ” part of the theorem follows immediately from Lemma 2.5. So let us turn to the “only if ” part . As sume that F is not blo ck-res p ecting, i.e. there is an I ∈ F ∗ and there is a blo c king D of I suc h that ev ery J ⊂ I satisfying (2 .1) is no t F -stationary . In other w ords N \ J ∈ F for ev ery J ⊂ I satisfying (2.1). Since the finite in tersection of the filter elemen ts again b elongs to F , w e can reform ulate the fact that F is not blo ck - resp ecting a s f ollo ws: there is an I ∈ F ∗ and suc h a blo c king D = { D k } k ∈ N of I that N \ J ∈ F for ev ery J ⊂ I satisfying (2.4) sup k ∈ N | J ∩ D k | < ∞ . No w, using Dv oretzky’s almost Euclidean section theorem let us select an increasing sequence of in tegers 0 = m 0 < m 1 < m 2 < . . . and a sequence of ve ctors x n ∈ ℓ 1 suc h that x n = 0 whe n n 6∈ I ; x n ∈ Lin { e k } k ∈ ( m i − 1 ,m i ] when n ∈ D i and for ev ery collection of scalars a n (2.5) X n ∈ D i | a n | 2 ! 1 / 2 ≤      X n ∈ D i a n x n      ≤ 2 X n ∈ D i | a n | 2 ! 1 / 2 . This sequence conv erges co ordinate-wise to 0 and is not F -conv ergen t to 0 in norm, b ecause k x n k ≥ 1 for ev ery n ∈ I . Let us prov e x n ’s w eak F -con v ergence to 0, whic h will show that F do es not hav e the simplified Sc h ur pro p ert y . W ell, take an f ∈ ℓ ∗ 1 with k f k = 1 , fix an ε > 0 and consider the set o f indexes A = { n : f ( x n ) < ε } . W e m ust prov e that A ∈ F . Since the complemen t of A lies in I , it is sufficien t t o sho w that J = N \ A = { n : f ( x n ) ≥ ε } satisfies (2.4). In other w ords w e m ust estimate d k = | J ∩ D k | from ab ov e uniformly in k . Let us do this. Consider y k = P n ∈ J ∩ D k x n . Then f ( y k ) ≥ εd k and due to (2.5) k y k k 2 ≤ 2 d k . Hence εd k ≤ f ( y k ) ≤ p 2 d k and d k ≤ 2 /ε 2 .  R emark 2.7 . One can see that in the “only if” part of t he Theorem 2.6 pro of t he sequence ( x n ) is b ounded by √ 2 . So, if one restricts Defini- tion 2.1 to the b ounded sequen ces, the class of filters do es not c hange. In f a ct this is a little bit surprising b ecause a we akly F -conv ergent 6 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV sequence can con v erge to infinit y in norm [6]. If one analyzes the c haracterization [6 ] o f those “go o d” filters F for whic h ev ery w eakly F - con v ergen t sequence has a norm-b ounded subsequen ce, one can see that ev ery simple Sc hur filter is “go o d”. The only obstacle to see this without refereeing to [6] is the co ordinate-wise con ve rgence whic h a p- p ears in Definition 2.1. T o see that this obstacle is not fa tal one really needs to go into the pro ofs of [6]. 3. Schur fil te rs Let us pass no w to the study of the most natural Sc h ur theorem gen- eralization, whic h is easier to f orm ulate, but is muc h more complicated to c haracterize in com binatorial terms. Definition 3.1. A filter F on N is said to b e a Schur filter (o r is said to hav e the Schur pr op erty ) if for eve ry w eakly F -conv ergent to 0 sequence ( x n ) ⊂ ℓ 1 , n ∈ N the F - lim k x n k equals 0. Eviden t ly , eve ry Sch ur filter has the simplified Sch ur prop ert y . By no w we do n’t kno w if the con ve rse holds true as w ell. T o simplify the exposition w e mostly consider N = N , but the general case cannot differ from this particular one. Definition 3.2. F is said to b e a diagonal filter if for ev ery decreasing sequence ( A n ) ⊂ F of the filter elemen ts a nd for ev ery I ∈ F ∗ there is a J ∈ F ∗ , J ⊂ I suc h that | J \ A n | < ∞ for all n ∈ N . Lemma 3.3. If a filter F on N is diago n al then for every I ∈ F ∗ and for every c o or din a te-wise F -c o nver gent to 0 s e quenc e ( x n ) ⊂ ℓ 1 ther e is a J ∈ F ∗ , J ⊂ I such that x n c o or d i n ate-wise c onver ge to 0 along J . Pr o of. F ix a decreasing sequence of subsets U n , fo rming a base o f neigh b orho o ds of 0 in the t op ology of co ordinate-wise con v ergence. Define A n = { k ∈ N : x k ∈ U n } . Since F is diagonal there is a J ∈ F ∗ , J ⊂ I suc h that | J \ A n | < ∞ for all n ∈ N . This is the J w e desire.  R emark 3 .4 . As one can see fro m the pro of the only pr o p ert y of the co ordinate-wise con v ergence t o p ology w e used is the countable base o f 0 neigh b orho o ds existence. Also one can easily prov e t he inv erse to the Lemma 3.3 r esult: if F is not diagonal, then there is a I ∈ F ∗ and a coo rdinate-wise F -conv ergent to 0 seq uence ( x n ) ⊂ ℓ 1 suc h that for ev ery J ∈ F ∗ , J ⊂ I the sequence ( x n ) do es not con v erge co ordinate-wise to 0 a long J . Let us demonstrate this in v erse theorem. By the negation of the diagonality definition a decreasing sequence of A n ∈ F and an I ∈ F ∗ exist such that if S ⊂ I satisfies the condition | S \ A n | < ∞ for a ll n ∈ N then N \ S ∈ F . Without loss of generality one ma y assume THE SCHU R ℓ 1 THEOREM FOR FIL TERS 7 that all the D n := A n \ A n +1 are infinite and S n D n = I . Then ev ery J ∈ F ∗ , J ⊂ I m ust satisfy condition (3.1) |{ n ∈ N : | J ∩ D n | = ∞}| = ∞ . F or ev ery n ∈ I denote b y f ( n ) suc h index that n ∈ D f ( n ) . Consider the following sequence ( x n ) : for n ∈ N \ I put x n = 0 , and for n ∈ I put x n = e n + e f ( n ) . This sequence is the one we need. Theorem 3.5. I f a filter F on N is diagonal and is blo ck-r es p e cting, then F has the Schur pr op erty. Pr o of. Let ( x n ) ⊂ ℓ 1 b e w eakly F -con vergen t to 0 . Arguing “ad absurdum” supp ose that there is an I ∈ F ∗ suc h tha t (3.2) inf n ∈ I k x n k > ε > 0 . Due to Lemma 3.3 there is a J ∈ F ∗ , J ⊂ I suc h that x n co ordinate- wise conv erge to 0 along J . Since F ( J ) is blo c k-resp ecting (Remark 2.4), the condition ( 3 .2) con tradicts Theorem 2.6.  It w as show n in Theorem 2.6 that the blo c k-resp ect of F is a nec- essary condition in order to b e a Sc h ur filter. O ur next go a l is to sho w that the diagonality of F is not a necessary conditio n. T o do this define a sp ecial filter on N . Let D = { D n } n ∈ N b e a disjoin t partitio n of N in to infinite subsets. F or ev ery sequence C = { C n } n ∈ N of finite subsets C n ⊂ D n and ev ery m ∈ N in tro duce the set B m,C = S ∞ n = m ( D n \ C n ) . The sets B m,C form a filter base. D enote the corresp onding filter b y F D . One can easily see that F D is a n example of not diagonal blo c k- resp ecting filter. In fa ct this filter “a lmost” app eared in Remark 3.4. T o make the picture clearer, w e ma y represen t N as an infinite matrix N × N , with D n = { ( k , n ) : k ∈ N } b eing its columns. Definition 3.6. A filter F on N is said to b e self-r epr o d ucing if for ev ery I ∈ F ∗ there is a J ∈ F ∗ , J ⊂ I such that the structure of the trace of F on J is the same as of the original filter F , i.e. there is a bijection s : N → J , that maps F into its tra ce on J : A ∈ F ⇐ ⇒ s ( A ) ∈ F ( J ) . Theorem 3.7. F D is a S c h ur filter, i.e. diag onality is not a ne c essa ry c ondi tion for the filter’s Schur pr op erty. Pr o of. F ir st remark, tha t a subset J ⊂ N is F D -stationary if and only if the condition (3.1) is met. In particular, for ev ery infinite subset M ⊂ N and for eve ry selection of infinite subsets A n ⊂ D n , n ∈ M the set S n ∈ M A n is a n F D -stationary set. Let us call suc h sets of the form S n ∈ M A n “standard sets”. Ev ery F D -stationary set con tains a standard subset. Remark also that the structure of the tra ce of F D on a standard subset J is exactly the same as of the origina l filter F D , i.e. F D is self-repro ducing. 8 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV T o prov e the theorem assum e con trary that there is a sequence ( x n ) ⊂ ℓ 1 , n ∈ N that F D -w eakly con v erge to zero but t he norms do not F D -con v erge to zero. So there is an ε > 0 and such a standard set J ⊂ N , that k x n k ≥ ε for a ll n ∈ J . According to the previ- ous remark, w e ma y a ssume without loss of generalit y that J = N , i.e. k x n k ≥ ε for a ll n ∈ N . Pass ing from x n to x n / k x n k we ma y supp ose that k x n k = 1 for all n . F or ev ery fixed m ∈ N se lect a subseque nce o f D ′ m ⊂ D m , suc h that x n , n ∈ D ′ m co ordinate-wise con v erge to an elemen t y m ∈ ℓ 1 . P assing to a new standard set of in- dexes S m ∈ N D ′ m w e reduce the situation to the case when x n , n ∈ D m con v erge co ordinate-wise t o y m for ev ery m ∈ N . Notice that due to the w eak F D -con v ergence t o zero o f the whole sequence ( x n ) , n ∈ N , y m con v erge co ordinate-wise to zero. In fact, for arbitrary co ordina t e functional e ∗ k and for ev ery ε > 0 t here is a set of the form B m,C suc h that | e ∗ k ( x j ) | < ε for all j ∈ B m,C . This means that for i ∈ N , i > m w e hav e | e ∗ k ( y i ) | = lim j ∈ D i | e ∗ k ( x j ) | ≤ ε. This means in its turn t he desired co ordinate-wise con vergenc e t o zero of ( y m ) . In tro duce a notatio n: f or n ∈ N denote by f ( n ) suc h index that n ∈ D f ( n ) . Put z n = x n − y f ( n ) . Consider t w o cases. T he first one: k z n k → F D 0 . In this case k y m k → 1 as m → ∞ , but on the other hand the condition y f ( n ) = x n − z n w → F D 0 implies ordinary w eak con v ergence of ( y m ) to 0, whic h is imp ossible according to the Sch ur theorem. In the remaining case, there is a standard set o n whic h k z n k are b ounded from b elow , so we may again without loss of generalit y assume that k z n k > ε > 0 for all n ∈ N . Claim. There is suc h a standard set J ⊂ N that the sequence ( z n ) n ∈ J is equiv alen t to the canonical basis of ℓ 1 . Pro of of the claim. Fix a decreasing sequence of δ k > 0 , k ∈ N , P k ∈ N δ k ≤ ε/ 8 . Using the definition of ℓ 1 let us select naturals m ( n ) suc h tha t f or ev ery n ∈ N the condition X k ≥ m ( n ) | e ∗ k ( z n ) | < δ n holds true. T a ke an arbitra ry n 1 ∈ D 1 . Now using consequen tly the co ordinate-wise con v ergence to 0 of sequences ( z n ) , n ∈ D m for v alues of m = 1 , 2 , 1 , 2 , 3 , 1 , 2 , 3 , 4 , . . . sele ct a sequence ( n i ) ⊂ N in suc h a w a y that n 2 ∈ D 1 , n 3 ∈ D 2 , n 4 ∈ D 1 , n 5 ∈ D 2 , n 6 ∈ D 3 , etc. (lik e triangle en umeration of a matrix) and for ev ery i ∈ N X k ≤ s ( i ) | e ∗ k ( z n i +1 ) | < ε 2 i +3 , THE SCHU R ℓ 1 THEOREM FOR FIL TERS 9 where s ( i ) denotes max k ≤ i m ( n k ) . Under this construction J = ( n i ) i ∈ N is a standard se t, and z n i is just a small p erturbation of the blo c k-basis w i = P k ∈ ( s ( i − 1) ,m ( n i )] e ∗ k ( z n i ) e k , whic h due to the Lemma 1.2 means tha t the claim is prov ed . No w the la st step. Once more without loss of generality assume that J ⊂ N from the Claim in fact equals N , i.e. ( z n ) n ∈ N are equiv alen t to the canonical basis of ℓ 1 . Then for ev ery b o unded seque nce of scalars ( a n ) n ∈ N there is a functional x ∗ ∈ ℓ ∗ 1 suc h that x ∗ ( z n ) = a n for all n ∈ N . Select these a n = ± 1 in suc h a w ay that for ev ery i ∈ N |{ n ∈ D i : a n = 1 }| = |{ n ∈ D i : a n = − 1 }| = ∞ . Then for the corr esp onding functional x ∗ w e hav e for ev ery i ∈ N lim sup n ∈ D i x ∗ ( x n ) − lim inf n ∈ D i x ∗ ( x n ) = lim sup n ∈ D i x ∗ ( z n ) − lim inf n ∈ D i x ∗ ( z n ) = 2 , whic h contradicts w eak F D -con v ergence of x n .  4. Ca tegor y respecting and strongl y dia gonal fil ters and ul trafil ters Let us in tro duce o ne more class o f filters, whic h are blo c k-resp ecting and diagonal at the same time. Definition 4.1. F is said to b e str ongly diag onal if fo r ev ery decreas- ing sequence ( A n ) ⊂ F of the filter elemen ts and f o r eve ry I ∈ F ∗ there is a J ∈ F ∗ , J ⊂ I suc h that (4.1) | ( J ∩ A n ) \ A n +1 | ≤ 1 for all n ∈ N . According to the Theorem 3.5 all strongly diagonal filters hav e the Sc h ur prop erty . Definition 4.2. A filter F o n N is said to b e c ate gory r es p e cting if for ev ery compact metric space K and fo r ev ery f amily of closed subsets ( F A ) A ∈F of K if F A ⊂ F B , whenev er B ⊂ A in F , and K = S A ∈F F A then in t( F B ) 6 = ∅ for some B ∈ F . The obv ious examples of category resp ecting filters are those of coun table base. Moreo v er, eve ry filter with a base of cardinality k < m is category respecting (see [5], p. 3-4 for the definition of m and The- orem 13A, p. 16 for the corresp onding result). But the Martin Axiom means that m equals the cardinality of contin uum, so if w e accept the Martin Axiom together with negatio n of the con tinuum h yp othesis, w e can go to some filters with uncoun table base. The pro of of Sc hur prop erty for ℓ 1 using the Baire theorem as pre- sen ted in [3, Prop ostion 5.2] give s a hin t that category resp ecting filters are related to the Sch ur prop ert y . The next theorem sho ws t ha t in fact 10 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV to b e category resp ecting is a stronger restriction than to hav e the Sc h ur prop erty . Theorem 4.3. If F is a c ate gory r esp e cting filter on N , then F is str ongly diag o n al. Pr o of. Assume con trary that F is not strongly diagonal, i.e. there is a decreasing sequence ( A n ) ⊂ F of the filter elemen ts and there is an I ∈ F ∗ suc h that for all J ∈ F ∗ , J ⊂ I the condition (4.1) is not met. Without loss of generalit y w e ma y a ssume that the filter is defined only on I (pass to the trace of F on I ), that T n ∈ N A n = ∅ (this in tersection is not statio na ry , so w e may just erase this interse ction from I ) a nd that all D n := A n \ A n +1 are not empt y . If one pic ks up a sequence of finite subsets (4.2) C n ⊂ D n , sup n ∈ N | C n | < ∞ then N \ [ n ∈ N C n ∈ F . Let us in tro duce the fo llowing compact top olog ical spaces e D n : if D n is finite then e D n = D n with discrete top o logy; if D n is infinite then e D n = D n S {∞ n } – the one-p oint compactification of D n . Recall t ha t K = Q n ∈ N e D n is compact in co ordinate-wise conv ergence top ology and metrizable. D efine a family of closed sets ( F A ) A ∈F in K as fo llows: F A = { x ∈ K : π n ( x ) ∈ e D n \ A for all n ∈ N } , where π n : K → e D n stands for the n - th co ordinate pro jection. These sets are closed and ha v e empt y in terior (the inte rio r could b e non-empty only if for a sufficien tly large m D n ∩ A = ∅ for all n ≥ m , whic h is not the case b ecause S k ≥ m D k = A m ∈ F ). F or ev ery x ∈ K the set A ( x ) = N \ S n ∈ N { π n ( x ) } is a filter elemen t ( due to ( 4 .2)) and x ∈ F A . So the union of all ( F A ) A ∈F equals K . Con tradiction.  Corollary 4.4. If F is a c ate gory r esp e cting filter on N , then F is a Schur filter. Corollary 4.5. Every filter wi th a c ountable b ase is str ongl y diag o nal. Theorem 4.6. Under the assumption of c ontinuum hyp othesis ther e i s a str ongly diagonal ultr afi lter. Pr o of. D enote b y Ω the set of all coun table ordinals. Let us en umerate as ( I ( α ) , A ( α ) ) , α ∈ Ω all the pairs ( I , A ) , where I is an infinite subset of N , and A is a decreasing sequenc e of infinite subsets of N : A ( α ) = ( A n ( α )) n ∈ N , N ⊃ A 1 ( α ) ⊃ A 2 ( α ) . . . . W e construct recurren tly an increasing family F α , α < ω 1 of filters with countable base and an increasing family of sets Ω α ⊂ Ω , as fo llo ws: F 1 is the F rec h´ et filter, Ω 1 = ∅ . If w e hav e a n ordinal of the fo rm α + 1 w e pro ceed as follo ws: w e find the smallest β ∈ Ω \ Ω α suc h that I ( β ) ∈ F ∗ α and suc h that A ( β ) consists of F α elemen t s. Applying THE SCHU R ℓ 1 THEOREM FOR FIL TERS 11 Corollary 4.5, we find a J ∈ F ∗ α , J ⊂ I = I ( β ) suc h that (4.1) holds true for A n = A n ( β ) . Define F α +1 as the filter generated b y F α and J , and put Ω α +1 = Ω α ∪ { β } . If α is a limiting ordinal, put F α = S β <α F β and Ω α = S β <α Ω β . Define the filter F w e need as F = S β <ω 1 F β . Let us demonstrate that F is an ultrafilter. T o do this w e m ust prov e that F ∗ ⊂ F . Let B ∈ F ∗ . Then B ∈ F ∗ α for all α . Let β ∈ Ω b e the smallest ordinal, for whic h I ( β ) = B and A ( β ) consists of filter F elemen ts. Then there is an α , for whic h all A n ( β ) b elong to F α . If β ∈ Ω α this means that the pair ( I ( β ) , A ( β )) has app eared in our recurren t construction, and a subset J of B (and hence B itself ) was added to the filter. If not, then not later than on the step α + 1 + β this pa ir ( I ( β ) , A ( β )) has app eared in our recurren t construction and a subset J of B w as added to the filter. By the same argumen t F is strong ly diagonal.  Notice that t he diagonalit y of an ultrafilter F is equiv alen t to the follo wing we ll- kno wn prop erty: F is a “P-p o in t of β N ” . The consis- tency of P- p oin ts non-existence is a celebrated result of Shelah [11]. So, since ev ery stro ng ly diag onal filt er is diag o nal some set theoretic assumption is needed for the last theorem. By the wa y in the setting of ultrafilters a prop ert y equiv alen t t o “blo ck - resp ect”, called “ Q -p oin t of β N ” w as a lso studied and the non-existence o f Q-p oin ts is also known to b e consisten t [9]. T o conclude this section let us presen t an example of a strongly diagonal filter which is not catego ry resp ecting. This example resem bles strongly the pro of of Theorem 4.3. Let D = { D n } n ∈ N b e a disjoin t partition of N in to infinite subsets . F or ev ery sequence C = { C n } n ∈ N of finite subsets C n ⊂ D n in tro duce the set B C = S n ∈ N ( D n \ C n ) . The sets B C form a filter base. D enote the corresponding filter b y F d . A set J ⊂ N is F d -stationary if and only if there is an n ∈ N s uch that | J ∩ D n | = ∞ . One can easily see that F d is strongly diago na l. T o sho w that it is not categor y resp ecting consider the same system of subsets ( F A ) A ∈F of the same compact K as in the pr o of of the Theorem 4 .3 . The only difference is that no w in the definition of K w e don’t need to consider the case o f finite D n . These sets F A are closed, they hav e empty in terior, but their union con tains all t he K , whic h would b e imp ossible if F d w as category r esp ecting. 5. Weak sequential completene ss t he orem for fil ters Definition 5.1. A filter F on N is said to b e we ak ℓ 1 c omp l e te (or in abbreviated f orm WC 1- filter) if for ev ery F -con ve rg en t in the top ology σ ( ℓ ∗∗ 1 , ℓ ∗ 1 ) b ounded sequence ( x n ) ⊂ ℓ 1 its weak * F -limit x ∈ ℓ ∗∗ 1 in fact b elongs to ℓ 1 . 12 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV It is known that eve ry Banac h space with the Sc h ur pro p ert y is w eakly sequen tially complete. The next theorem together with the Theorem 4.6 show s that the picture for filters is more colorful. Theorem 5.2. A n ultr afilter c ann ot b e we ak ℓ 1 c omp l e te. Pr o of. Let F b e a (free as alw ays) ultrafilt er o n N . Consider an arbitrary f = ( f 1 , f 2 , . . . ) ∈ ℓ ∞ = ℓ ∗ 1 . Then for the canonical basis ( e n ) of ℓ 1 w e hav e lim F f ( e n ) = lim F f n , whic h sho ws tha t the sequence ( e n ) w eakly* F -conv erges to the func- tional lim F on ℓ ∞ , whic h eviden tly do es not b elong to ℓ 1 .  T o show that a W C1-filter may ha v e no Sc h ur pro p ert y (and eve n t o b e without t he simplified Sc hu r pr o p ert y), let us recall some elemen ts of statistical con vergenc e theory [4], [2]. A sequence ( x k ) in a top ological space X is statistic al ly c onver gent to x if for ev ery neighborho o d U of x the set { k : x k 6∈ U } has natural density 0, where the natural densit y of a subset A ⊂ N is defined to b e δ ( A ) := lim n n − 1 |{ k ≤ n : k ∈ A }| . Denote F s = { I ⊂ N : δ ( N \ I ) = 0 } and r emark that F s is a filter. As it is easy to see, F s -con v ergence and statistical con ve rg ence coincide, and a set J is F s -stationary prov ided δ ( J ) 6 = 0 . Recall that a scalar sequence ( x k ) is said to b e strongly Ces a r o - summable if there is a scalar x suc h that lim n →∞ 1 n n X j = 1 | x − x j | = 0 . It is kno wn that a b ounded scalar sequence is statistically con v ergent if and only if it is strongly Cesaro-summable (f o r a general v ersion of this criterion see [1 , Theorem 8]). Let us apply this fact. Theorem 5.3. F s is a WC1-filter but do es not h ave the s i m plifie d Schur pr op erty. Pr o of. Consider the blo cking of N in to D n = (2 n − 1 , 2 n +1 ] . Ev ery set J ⊂ N in tersecting eac h of D n b y no more than one elemen t, has zero natural densit y and consequen tly cannot b e F s -stationary . Hence F s is not blo c k-resp ecting and by t he Theorem 2.6 F s do es not ha v e the simplified Sc hu r prop erty . Let us show no w the we ak ℓ 1 completeness of F s . Let ( x n ) ⊂ ℓ 1 b e a b ounded sequence and let weak* F s -limit of ( x n ) b e equal to an x ∗∗ ∈ ℓ ∗∗ 1 . This means that for ev ery f ∈ ℓ ∗ 1 lim n →∞ 1 n n X j = 1 | f ( x ∗∗ − x j ) | = 0 . THE SCHU R ℓ 1 THEOREM FOR FIL TERS 13 Hence the v ectors 1 n P n j = 1 x j w eakly* con v erge to x ∗∗ as n → ∞ . By the ordinary w eak sequen tial completeness of ℓ 1 this means that x ∗∗ ∈ ℓ 1 .  Our next goal is to sho w that if one av o ids ultra filters in a reasonable sense, then the same sufficien t condition whic h we ha ve for the Sch ur prop ert y works for the W C1 as w ell. Definition 5.4. A filter F on N is said to b e a p ap er filter ( p-filter ) if all the tra ces o f F o n F -stat io nary subsets are not ultrafilters. Theorem 5.5. I f a p-filter F on N is diagonal and is blo ck- r esp e cting then F is a WC1-filter. Pr o of. Let ( x n ) ⊂ ℓ 1 b e a b ounded sequenc e and let F -limit of ( x n ) in the top ology σ ( ℓ ∗∗ 1 , ℓ ∗ 1 ) b e equal to an x ∗∗ ∈ ℓ ∗∗ 1 \ ℓ 1 . Consider the standard pro jection P : ℓ ∗∗ 1 → ℓ 1 , whic h maps eve ry elemen t of ℓ ∗∗ 1 (i.e. a linear f unctional on ℓ ∞ ) into its restriction on c 0 . Denote x = P x ∗∗ . Without loss of generalit y we may assume that x = 0 : otherwise consider x n − x instead of x n . This assumption means that x n co ordinate-wise con v erge to 0 with resp ect to the filter F . Due to the Lemma 3.3 there is a I ∈ F ∗ , suc h that x n co ordinate-wise con v erge to 0 a lo ng I . Since F ( I ) is blo ck-respecting (Remark 2.4), w e ma y apply Lemma 2.5 to get suc h a J ∈ F ∗ J ⊂ I , that the sequence ( x n ) , n ∈ J is equiv alen t to the canonical basis of ℓ 1 (here w e use also the b oundedness of the sequenc e). Since F ( J ) is no t a n ultrafilter we can decomp ose J in to tw o disjoin t F -stationary subsets J 1 and J 2 . Consider a functional x ∗ ∈ ℓ ∗ 1 whic h tak es v a lue 1 on all x n , n ∈ J 1 and is equal to − 1 o n ev ery x n , n ∈ J 2 . Then 1 = lim F ( J 1 ) x ∗ ( x n ) = x ∗ ( x ∗∗ ) = lim F ( J 2 ) x ∗ ( x n ) = − 1 . This con tra diction completes the pro of.  T o pro ceed further let us introduce the sum and the pro duct of filters. Definition 5.6. Let F 1 , F 2 b e filters on N 1 and N 2 resp ectiv ely . Define F 1 + F 2 as the filter on N 1 ∪ N 2 consisting of those elemen ts A ⊂ N 1 ∪ N 2 that A ∩ N 1 ∈ F 1 and A ∩ N 2 ∈ F 2 . The filter F 1 × F 2 is defined on N 1 × N 2 with ba se f ormed by the sets A 1 × A 2 , A 1 ∈ F 1 , A 2 ∈ F 2 . Definition 5.7. A filter F o n N is said to ha ve the double Sc h ur pr op erty if F × F is a Sc h ur filter. Theorem 5.8. Every filter F with the double S chur pr op erty is a WC1-filter and a Schur filter at the same time. Pr o of. Consider such a b ounded sequence ( x n ) ⊂ ℓ 1 that F -limit of ( x n ) in the top olo gy σ ( ℓ ∗∗ 1 , ℓ ∗ 1 ) is equal to an x ∗∗ ∈ ℓ ∗∗ 1 . Then the double sequence ( x n − x m ) is w eakly F × F -conv ergent to 0 . According 14 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV to the double Sch ur prop erty of F this implies that k x n − x m k → F ×F 0 , i.e. (due to the completeness of ℓ 1 ) there is an elemen t x ∈ ℓ 1 suc h that k x n − x k → F 0 . Evide ntly x ∗∗ = x ∈ ℓ 1 .  6. Domina tion by Schur and WC1 fil ters. Op en problems Definition 6.1. A prop ert y P of filt ers (or corresponding class of filters) is said to b e quasi-incr e asing if for ev ery F ∈ P all the filters of the form F ( J ) for ev ery J ∈ F ∗ ha v e the prop erty P a s w ell. R emark 6.2 . F ( J ) - conv ergence to 0 (in arbitr a ry fixed to p ology) o f a sequence ( x n ) is equiv a len t to F -conv ergence to 0 in the same top ol- ogy o f the sequen ce ( x n χ J ( n )) . Consequen tly the prop erties defined only throug h conv ergence to 0 (lik e Sch ur or double Sc h ur prop erties) are quasi-increasing. Definition 6.3. A prop erty P of filters is said to b e d e cr e asi n g if for ev ery F ∈ P all the filters dominated b y F ha v e the prop ert y P as w ell. Eviden t ly W C1 filters fo r m a decreasing class. So one can impro ve the Theorem 5.8 as follo ws: ev ery filter dominated by a filter with the double Sc h ur prop ert y is a W C1-filter. This is an impro v emen t, b ecause of the fo llo wing prop osition: Theorem 6.4. The S chur pr op erty, the double Schur pr op erty a n d mor e over eve ry non-trivial quasi-incr e asing pr o p erty P of filters ar e not de cr e asing. Pr o of. Let F 1 ∈ P , F 2 6∈ P b e filters on N 1 and N 2 resp ectiv ely . Then F = F 1 + F 2 is a filter on N 1 ∪ N 2 whic h cannot ha v e the prop ert y P , b ecause F ( N 2 ) 6∈ P . On the other hand F ( N 1 ) ∈ P but F ( N 1 ) dominates F .  One can intro duce a bit w eak er but still reasonable v ersion of the Sc h ur prop erty , whic h is decreasing: Definition 6.5. A filter F on N is said to b e an almost Schur filter (or is said to ha v e the almost Sch ur pr op erty ) if fo r ev ery w eakly F - con v ergent to 0 sequence ( x n ) ⊂ ℓ 1 , n ∈ N the norms of x n are not separated from 0 (or in other w ords 0 is a cluster p oint for k x n k , n ∈ N ). Theorem 1.1 easily implies that a filter F on N has the Sc h ur prop ert y if and only if all the filters F ( J ) for ev ery J ∈ F ∗ are almost Sc hur filter. One can also introduce increasing prop erties: Definition 6.6. A prop erty P of filters is said to b e inc r e asing if for ev ery F ∈ P all the filters that dominate F ha v e the prop erty P as w ell. THE SCHU R ℓ 1 THEOREM FOR FIL TERS 15 Eviden t ly the negation of a n increasing prop ert y is a decreasing one and con tra v ersa. Definition 6.7. Let P b e an increasing (decreasing) class of filters. A class of filters P 1 ⊂ P is said to b e a b asis for P if P is the smallest increasing (decreasing) class, containing P 1 . The problem whic h lo oks in teresting is to construct explicitly a class of almost Sc h ur filters, whic h forms a base for the class of all almost Sc h ur filters. The same question mak es sense fo r the negation of prop- ert y to b e almost Sc h ur filter. Suc h a study w as done in [6] fo r the class o f those filters F , that w eak F -con ve rgence o f a sequence im- plies existence of a b ounded subsequence. Reference s [1] Connor, J. Two v alued measures and s ummabilit y , Analysis 10 (1990 ), 37 3- 385. [2] Connor, J. A top ologica l and functiona l analytic approach to sta tistical co n- vergence, Analy sis of Div er gence (Orono , ME , 1 997), Applied Numer. Harmon. Anal., Birkhauser , Bosto n, MA, 1999, 403- 413. [3] J. B. Conw ay , A c ourse in functional analysis , Gradua te T exts in Mathematics, vol. 96, Springer -V er lag, New Y or k , 1985. MR 86h:460 0 1 [4] F a st, H., Sur la convergence statistique, Collo q . Math. 2 (1951 ), 241 -244. [5] F r emlin, D. H., Cons equences of Martin’s axiom, Cambridge T racts in Math- ematics, vol. 84, Cambridge Universit y Pr ess, Ca m bridg e 198 4 (MR780 933 (86i:030 01)). [6] Ganichev M., Kadets V. Filter Convergence in Banach Spaces and gener alized Bases / in T aras Banakh (editor ) General T op ology in Banach Space s : NOV A Science Publishers, Huntington, New Y or k, 2001; pp. 61 - 6 9. [7] Kadets V., Leo nov A. Dominated conv er g ence and Egorov theo rems for filter conv ergenc e , Mathematical Physics, Analy sis, Ge o metry (MA G), vol. 3 (2 007), no. 2, 196- 212. [8] Lindenstrauss J., Tzafrir i L. Cla ssical Banach Spaces I and I I / Springer - V er lag, Berlin, 1996 . [9] Miller, Arnold W. Ther e a re no Q -p oints in Lav er’s mo del for the Bo rel con- jecture. Pro c. Amer. Math. So c. 78 (1980), no. 1, 10 3-106 . [10] T o dor cevic, S. T opics in top olog y . Lecture Notes in Ma thematics 1652 , Springer V er lag, Berlin, 19 97. [11] Wimmers, Edward L. The Shela h P -p o int indep endence theorem. Isra el J. Math. 43 (1982 ), no. 1, 28–48 . Equipe de Logique Ma th ´ ema tique UFR d e Ma th ´ ema tiques (case 70 1 2) Universit ´ e Denis-Dider o t P aris 7 2 place Ju ssieu 75251 P aris Cedex 05 France E-mail addr ess : av ileslo @um.es Dep ar t amento de Ma temticas, U niversidad de Murcia, 30100 E s- pinardo, Mur cia, Sp ain E-mail addr ess : be ca@um. es 16 A. A VILES, B. CASCALES, V. KADETS AND A . LEONOV Dep ar tment of Mechanics and Ma thema tics, Kharkov N a tional Uni- versity, pl. Svobody 4, 61077 Kharko v, Ukraine E-mail addr ess : vo va1kad ets@ya hoo.com Dep ar tment of Mechanics and Ma thema tics, Kharkov N a tional Uni- versity, pl. Svobody 4, 61077 Kharko v, Ukraine E-mail addr ess : le onov family @mail. ru