A strong boundedness result for separable Rosenthal compacta

A STR ONG BOUNDEDNESS RESUL T F OR SEP ARABLE R OSENTHAL COMP ACT A P ANDELIS DODOS Abstract. It is prov ed that the c lass of separable Rosent hal compacta on the Can tor set ha ving a uniformly bounded dense sequenc e of cont inuous functions, is s trongly b ounded. 1. Introduction Our main result is a strong b oundedness result for the class o f separable Ro s en- thal compacta (that is, separable compact subsets o f the first Baire class – see [ADK] and [Ro2]) on the Cantor s et having a un iformly b ounded dense seq ue nc e of con tinuous fu nctions. W e shall deno te this class b y SR C. The phenomenon of strong bo undedness, which w as first touched b y A. S. K echris and W. H. W o o din in [KW], is a streng thening of the clas sical prop erty of b o undedness of Π 1 1 -ranks. Abstractly , one has a Π 1 1 set B , a natura l notion o f em b edding be tw een elements of B and a cano nical Π 1 1 -rank φ on B which is coherent with the embedding, in the sense that if x, y ∈ B and x embeds in to y , then φ ( x ) ≤ φ ( y ). The strong b ounded- ness o f B is the fact that for every analytic subset A of B there exists y ∈ B such that x embeds into y for every x ∈ A . Basic examples o f strong ly b ounded classes are the well-orderings WO and the well-founded trees WF (althoug h, in these cases strong b oundedness is easily seen to b e equiv alent to b oundednes s ). Recently , it was shown (see [AD] and [DF]) that several class es of se pa rable Banach spaces are strongly bo unded, where the corr esp onding notion o f em b edding is that of (linear) isomorphic em b edding. These results hav e, in turn, imp ortant co nsequences in the study of universality problems in Banach s pa ce Theory . W e will add another example to the list of s trongly b ounded classe s , namely the class SR C. W e notice that ev ery K in SRC ca n be naturally co ded by its dens e sequence of contin uous functions. Hence, we iden tify SRC with the se t  ( f n ) ∈ B (2 N ) N : { f n } p ⊆ B 1 (2 N ) and f n 6 = f m if n 6 = m  where B (2 N ) stands for the c lo sed unit ball of the separ able Ba nach space C (2 N ). With this iden tification, the set SRC is Π 1 1 -true. A cano nical Π 1 1 -rank on SRC comes from the work of H. P . Ro senthal. Specifically , for e very f = ( f n ) in SRC 1 2000 Mathematics Subje ct Classific ation : 03E15, 26A21, 54H05. 2 Key wor ds : separable Rosen thal compacta, s trongly b ounded classes. 1 2 P ANDELIS DODOS one is lo o k ing at the order of the ℓ 1 -tree of the sequence ( f n ). O ne ha s also a natural notion of top olog ical em b edding b etw een element s of SRC. In particular, if f = ( f n ) a nd g = ( g n ) a re in SRC, then we say that g top olo gic al ly emb e ds in to f , if ther e exis ts a homeomorphic em b edding of the compact { g n } p int o { f n } p . This top ologica l embedding, how ever, is r a ther w eak and not coher ent with the Π 1 1 -rank on SRC. Thu s, we streng then the notion of em b edding by impos ing e x tra metric conditions on the relation b et ween g and f . T o motiv a te o ur definition, ass ume that g = ( g n ) and f = ( f n ) were in addition Schauder basic sequences. In this case the most natura l thing to consider is equiv a le nce of basic sequenc e s, i.e. g embeds int o f if there ex ists L = { l 0 < l 1 < ... } ∈ [ N ] such that ( g n ) is equiv alen t to ( f l n ). In such a case, it is eas ily seen the o rder of the ℓ 1 -tree of g is dominated b y the one of f . Although not every sequence f ∈ SRC is Schauder basic, the following condition incorp ora tes the ab ov e observ ation. So, we say that g = ( g n ) str ongly emb e ds into f = ( f n ), if g top olog ically em b eds into f and, mo r eov er, for every ε > 0 there exists L ε = { l 0 < l 1 < ... } ∈ [ N ] such that for every k ∈ N and every a 0 , ..., a k ∈ R we hav e    max 0 ≤ i ≤ k   i X n =0 a n g n   ∞ −   k X n =0 a n f l n   ∞    ≤ ε k X n =0 | a n | 2 n +1 . The notion of strong embedding is co herent with the Π 1 1 -rank on SRC a nd is consis- ten t with our motiv ating obs e r v a tio n, in the sense that if g = ( g n ) strongly embeds int o f = ( f n ) a nd ( g n ) is Schauder basic, then ther e exists L = { l 0 < l 1 < ... } ∈ [ N ] such that ( f l n ) is Schauder basic and equiv alen t to ( g n ). Under the ab ove termi- nology , w e prove the following. Main Theo rem. L et A b e an analytic su bset of SRC . Then ther e exists f ∈ SRC such t hat for every g ∈ A the se quenc e g str ongly emb e ds into f . 2. Ba ckgr ound ma terial W e let N = { 0 , 1 , 2 , ... } . B y [ N ] we denote the set of a ll infinite subsets of N , while for every L ∈ [ N ] by [ L ] we denote the set of a ll infinite subsets o f L . F or every Polish space X by B 1 ( X ) w e denote the set of all real-v alued, Baire-1 functions on X . If F is a subset of R X , then by F p we shall denote the clos ure of F in R X . Our descriptive set theoretic nota tio n and terminolog y follows [Ke]. If X , Y are Polish spaces, A ⊆ X a nd B ⊆ Y , then we s ay that A is Wadge (resp ectively Bor el ) r e ducible to B if there exists a contin uous (resp ectively Bore l) map f : X → Y such that f − 1 ( B ) = A . If A is Π 1 1 , then a map φ : A → ω 1 is said to b e a Π 1 1 - r ank on A if there exist relations ≤ Σ , ≤ Π in Σ 1 1 and Π 1 1 resp ectively such that for all y ∈ A we hav e x ∈ A and φ ( x ) ≤ φ ( y ) ⇔ x ≤ Σ y ⇔ x ≤ Π y . A S TR ONG BOUNDEDNESS RESUL T 3 W e notice that if B is Bo rel reducible to a set A via a Borel map f and φ is a Π 1 1 -rank o n A , then the ma p ψ : B → ω 1 defined by ψ ( y ) = φ  f ( x )  for a ll y ∈ B is a Π 1 1 -rank o n B . 2.1. T rees. Let Λ b e a non-empty set. By Λ < N we denote the set of all finite sequences o f Λ. W e vie w Λ < N as a tre e equipp ed with the (str ic t) par tial or de r ⊏ of end-extension. If t ∈ Λ < N , then the length | t | of t is defined to be the cardina lity of the set { s ∈ Λ < N : s ⊏ t } . If s, t ∈ Λ < N , then by s a t we denote their concatenatio n. Two node s s, t ∈ Λ < N are said to b e c omp ar able if either s ⊑ t o r t ⊑ s ; otherwis e are said to b e inc omp ar able . A subset of Λ < N consisting of pairwise comparable no des is said to be a chain . If Λ = N and L ∈ [ N ], then by FIN( L ) we denote the subset of L < N consisting o f all finite strictly incr e asing seq ue nc e s in L . F or every x ∈ Λ N and every n ≥ 1 we set x | n =  x (0) , ..., x ( n − 1 )  ∈ Λ < N while x | 0 = ∅ . A tr e e T o n Λ is a downw a rds closed subset of Λ < N . By T r(Λ) we denote the set of all trees o n Λ. Hence T ∈ T r(Λ) ⇔ ∀ s, t ∈ Λ < N ( t ∈ T ∧ s ⊑ t ⇒ s ∈ T ) . A tree T on Λ is s aid to be prune d if for every t ∈ T there e xists s ∈ T with t ⊏ s . If T ∈ T r(Λ), then the b o dy [ T ] of T is defined to b e the set { x ∈ Λ N : x | n ∈ T ∀ n } . A tree T is said to b e wel l-founde d if [ T ] = ∅ . The subset of T r(Λ) co nsisting of all well-founded tree s on Λ will be denoted by WF(Λ). If T ∈ WF(Λ), we let T ′ = { t : ∃ s ∈ T with t ⊏ s } ∈ WF(Λ). By transfinite recursion we define the iterated deriv atives T ( ξ ) of T . The or der o ( T ) of T is defined to b e the least or dinal ξ suc h that T ( ξ ) = ∅ . If S, T ar e w ell-founded trees, then a map φ : S → T is called monotone if s 1 ⊏ s 2 in S implies that φ ( s 1 ) ⊏ φ ( s 2 ) in T . No tice that in this ca s e o ( S ) ≤ o ( T ). If Λ , M are non-empty s ets, then we identify every tree T on Λ × M with the set of all pa irs ( s, t ) ∈ Λ < N × M < N such that | s | = | t | = k and  ( s (0) , t (0)) , ...., ( s ( k − 1) , t ( k − 1))  ∈ T . If Λ = N , then w e shall simply denote by T r and WF the sets of all trees and well-founded tre e s on N respectively . F or every countable set Λ the set WF(Λ) is Π 1 1 -complete and the map T → o ( T ) is a Π 1 1 -rank on WF(Λ) (see [Ke]). 2.2. Sc hauder basic s equences. A sequence ( x n ) o f non-zer o vectors in a B a nach space X is said to b e a Schauder b asic se quenc e if it is a Schauder bas is o f its closed linear span (see [L T]). This is e q uiv alent to s ay that there exists a constant K ≥ 1 such that for every m, k ∈ N with m < k and every a 0 , ..., a k ∈ R w e hav e (1)   m X n =0 a n x n   ≤ K   k X n =0 a n x n   . The least cons tant K for which inequality (1 ) ho lds is called the b asis c onstant of ( x n ). A Schauder basic sequence ( x n ) is said to b e monotone if K = 1. It is said to be seminormalize d (resp ectively normalize d ) if ther e ex ists M > 0 s uch that 1 M ≤ k x n k ≤ M (res pe ctively k x n k = 1) for every n ∈ N . 4 P ANDELIS DODOS Let X and Y b e Banach spaces. If ( x n ) and ( y n ) a re t wo sequences in X and Y resp ectively and C ≥ 1, then we say that ( x n ) is C - e quivalent to ( y n ) (or simply equiv ale nt, if C is understo o d) if for every k ∈ N and every a 0 , ..., a k ∈ R w e hav e 1 C   k X n =0 a n y n   Y ≤ k k X n =0 a n x n   X ≤ C   k X n =0 a n y n   Y . W e deno te by ( x n ) C ∼ ( y n ) the fact that ( x n ) is C -equiv a lent to ( y n ). 3. Coding SR C Let X b e a compact metrizable space and let SRC( X ) be the family of all separable Rosenthal compacta on X having a de ns e set of co ntin uo us functions which is uniformly b ounded with r esp ect to the supr e m um no r m. W e denote by B ( X ) the c losed unit ball of the separa ble Banach space C ( X ). Notice that every K ∈ SR C( X ) is natura lly co ded by its dense sequence of con tin uous functions. Hence we ma y ident ify SR C( X ) with the set  ( f n ) ∈ B ( X ) N : { f n } p ⊆ B 1 ( X ) and f n 6 = f m if n 6 = m  . Let us denote b y B ( X ) the G δ subset of B ( X ) N consisting of all sequences f = ( f n ) in B ( X ) N such that f n 6 = f m if n 6 = m . With the ab ove identification the set SRC( X ) becomes a subset of the Polish spa ce B ( X ). Moreover, a s for every compact metrizable space X the Banach space C ( X ) embeds isometrically in to C (2 N ), we shall denote by SRC the set SRC(2 N ) a nd w e v iew SRC as the set o f all sepa rable Rosenthal compacta having a uniformly b ounded dense sequenc e of contin uous functions and defined on a compact metrizable space (it is crucial that C ( X ) embeds isometr ically in to C (2 N ) – this will b e clear later o n). The following lemma provides a n estimate for the complexity of the se t SR C( X ). Lemma 1. F or every c omp act met rizable sp ac e X t he set SRC( X ) is Π 1 1 . Mor e- over, the set SRC is Π 1 1 -true. Pr o of. Instead of calculating the c omplexity o f SRC( X ) we will a c tually find a Borel map Φ : B ( X ) → T r such that Φ − 1 (WF) = SR C( X ). In other words, we will find a Borel reduction of SR C( X ) to WF. This will not only show that SRC( X ) is Π 1 1 , but also , it will provide a natural Π 1 1 -rank o n SRC( X ). This ca nonical r e ductio n comes from the work of H. P . Rosenthal. Spec ific a lly , let ( e i ) b e the sta ndard ba sis o f ℓ 1 . F or every d ∈ N with d ≥ 1 and every f = ( f n ) in B ( X ) we asso ciate a tree T d f on N defined by s ∈ T d f ⇔ s = ( n 0 < ... < n k ) ∈ FIN( N ) and ( e i ) k i =0 d ∼ ( f n i ) k i =0 . Notice that ( e i ) k i =0 d ∼ ( f n i ) k i =0 if for every a 0 , ..., a k ∈ R w e hav e 1 d k X i =0 | a i | ≤   k X i =0 a i f n i   ∞ ≤ d k X i =0 | a i | . A S TR ONG BOUNDEDNESS RESUL T 5 Observe that for every t ∈ N < N the set { f : t ∈ T d f } is a closed subse t of B ( X ). This yields that the map B ( X ) ∋ f 7→ T d f ∈ T r is Borel (actually it is Baire-1). Next we glue the seq uenc e of tr ees { T d f : d ≥ 1 } and we o btain a tree T f on N defined b y the rule s ∈ T f ⇔ ∃ d ≥ 1 ∃ s ′ with s = d a s ′ and s ′ ∈ T d f . The tree T f is usually called the ℓ 1 -tree of the sequence f = ( f n ). Clea r ly the map Φ : B ( X ) → T r defined by Φ( f ) = T f is Bor el. W e o bserve that f = ( f n ) ∈ SRC( X ) ⇔ T f ∈ WF . This equiv alence is essentially Rosenthal’s Dichotom y [Ro1] (see also [Ke] and [T o]). Indeed, let f = ( f n ) b e such that T f is well-founded. By Ros ent hal’s Dichotom y , every subseq ue nc e of ( f n ) has a further p oint wise conv ergent subseq ue nc e . By the Main Theorem in [Ro2], the clos ure o f { f n } in R X is in B 1 ( X ), and so, f ∈ SRC( X ). Conv ersely assume that T f is ill-founded. Ther e exists L = { l 0 < l 1 < .. } ∈ [ N ] such that the sequence ( f l n ) is equiv ale nt to the standa rd basis of ℓ 1 . By the fa c t that ( f n ) is uniformly bo unded and Leb esg ue’s dominated co nv ergence theorem, we g et that the sequence ( f l n ) has no p oint wise conv ergent subseque nc e . This implies that the clo s ure o f { f n } in R X contains a homeomo rphic co py of β N , a nd so, f / ∈ SRC( X ). It follows that the map Φ determines a Borel reduction of SRC( X ) to WF . Hence the set SRC ( X ) is Π 1 1 and that the map φ X : SRC( X ) → ω 1 defined by φ X ( f ) = o ( T f ) is a Π 1 1 -rank on SRC( X ). W e pro ceed to show that the set SRC is Π 1 1 -true. Denote b y φ the canonical Π 1 1 -rank φ 2 N on SR C defined ab ov e. In order to pro ve that SRC is Π 1 1 -true, b y [Ke, Theorem 35.2 3], it is enough to show that sup { φ ( f ) : f ∈ SRC } = ω 1 . In the argument b elow w e s ha ll use the following simple fact. F act 2. L et X , Y b e c omp act metrizable s p ac es and e : X → Y a c ontinuous onto map. L et f = ( f n ) ∈ SRC( Y ) and define g = ( g n ) ∈ C ( X ) N by g n ( x ) = f n ( e ( x )) for every x ∈ X and every n ∈ N . Then g ∈ SR C( X ) and φ Y ( f ) = φ X ( g ) . Now let F b e a family of finite subsets of N which is heredita ry (i.e. if F ∈ F and G ⊆ F , then G ∈ F ) and compact in the p o int wis e topolo g y (i.e. compact in 2 N ). T o every such family F one as s o ciates its order o ( F ), which is s imply the order of the downw ards closed, well-founded tr ee T F on N defined by s ∈ T F ⇔ s = ( n 0 < ... < n k ) ∈ FIN( N ) and { n 0 , ..., n k } ∈ F . Such families are well-studied in Co m binatoric s and F unctional Analysis a nd a detailed expos itio n can be found in [A T]. What w e need is the s imple fact that for ev ery countable ordina l ξ o ne can find a compact hereditary family F with o ( F ) ≥ ξ . So, fix a count able o rdinal ξ and let F be a compact her e ditary family with o ( F ) ≥ ξ . W e will additionally assume that { n } ∈ F for all n ∈ N . Define 6 P ANDELIS DODOS π F n : F → R by π F n ( F ) = χ F ( n ) for a ll F ∈ F . Clearly for every n ∈ N we have π F n ∈ C ( F ) a nd k π F n k ∞ = 1 . Moreover, a s the family F co nt ains all singletons, we get π F n 6 = π F m if n 6 = m . It is ea sy to see that the seq ue nc e ( π F n ) conv erges p oint wise to 0, and so, ( π F n ) ∈ SRC( F ). Claim 3. We have that φ F  ( π F n )  ≥ o ( F ) ≥ ξ . Pr o of of Claim 3 . The pr o of is essentially ba sed on the fact that F is hereditary . Indeed, notice that if F = { n 0 < ... < n k } ∈ F , then ( e i ) k i =0 2 ∼ ( π F n i ) k i =0 or equiv ale ntly F ∈ T 2 ( π F n ) . T o see this, fix F = { n 0 < ... < n k } ∈ F and let a 0 , ..., a k ∈ R arbitrary . W e s e t I + =  i ∈ { 0 , ..., k } : a i ≥ 0  and I − = { 0 , ..., k } \ I + . Then, either P i ∈ I + a i ≥ 1 2 P k i =0 | a i | or − P i ∈ I − a i ≥ 1 2 P k i =0 | a i | . Assume that the second case o ccur s (the argument is symmetric). Let F − = { n i : i ∈ I − } ⊆ F ∈ F . Then F − ∈ F as F is here dita ry . Now obser ve that 1 2 k X i =0 | a i | ≤ − X i ∈ I − a i =   k X i =0 a i π F n i ( F − )   ≤   k X i =0 a i π F n i   ∞ ≤ 2 k X i =0 | a i | . It follows b y the ab ov e discussion that the identit y map Id : T F → T 2 ( π F n ) is a well- defined monotone map. The claim is prov ed. ♦ By F act 2 and Claim 3, we conclude that sup { φ ( f ) : f ∈ SR C } = ω 1 and the ent ire pro of is completed.  4. Topological a n d strong embedding Consider the classes SRC( X ) and SR C( Y ), where X and Y ar e compact metriz- able s paces, as they were co ded in the pre v ious section. There is a canonical notion of embedding b etw een elements of SR C( X ) a nd SRC( Y ) defined as follows. Definition 4. L et X , Y b e c omp act metrizable sp ac es, f = ( f n ) ∈ SRC( X ) and g = ( g n ) ∈ SRC( Y ) . We say t hat g top olo gic al ly emb e ds into f , in symb ols g < f , if ther e ex ists a home omorphic emb e dding of { g n } p into { f n } p . Clearly the notion of topo logical embedding is natural and meaningful, as f 1 < f 2 and f 2 < f 3 imply that f 1 < f 3 . How ever, in this s e tting, one als o has a canonical Π 1 1 -rank o n SRC and any no tion of embedding betw een elemen ts of SRC should be coher ent with this rank, in the sense that if g < f , then φ Y ( g ) ≤ φ X ( f ). Unfortunately , the top ologica l em b edding is not stro ng enoug h in or der to hav e this prop erty . Example 1. Let F 1 and F 2 be t w o compact hereditary families o f finite subsets of N . As in the proo f of Lemma 1, co nsider the sequences ( π F 1 n ) ∈ SRC( F 1 ) and ( π F 2 n ) ∈ SRC ( F 2 ). Both of them are p oint wise conv ergent to 0 . Hence, they ar e A S TR ONG BOUNDEDNESS RESUL T 7 top ologica lly eq uiv alent and clear ly bi-embedable. How ever, it is easy to s ee that the co rresp onding r anks o f the t wo sequences dep end only on the order of the families F 1 and F 2 , and so , they ar e totally unrela ted. W e are go ing to strengthen the notion o f top olog ical embedding betw een the elements o f SRC. T o motiv ate our definition, let f = ( f n ) , g = ( g n ) ∈ SR C and assume that b oth ( f n ) and ( g n ) are Schauder bas ic s e quences. In this c ase, the most natur al notion of em b edding is that of equiv a lence, i.e. g em b e ds in to f if there exists L = { l 0 < l 1 < ... } ∈ [ N ] such that the sequence ( g n ) is equiv alent to ( f l n ). It is easy to verify that, in this case, we do hav e that φ ( g ) ≤ φ ( f ). Although not every f ∈ SRC is a Schauder ba sic se q uence, there is a metric relation we can impo se on f and g whic h inco rp orates the above obser v ation. Definition 5. L et X , Y b e c omp act metrizable sp ac es, f = ( f n ) ∈ SRC( X ) and g = ( g n ) ∈ SRC( Y ) . We say that g str ongly emb e ds into f , in symb ols g ≺ f , if g t op olo gic al ly emb e ds into f and, mor e over, if for every ε > 0 ther e exists L ε = { l 0 < l 1 < ... } ∈ [ N ] such t hat for every k ∈ N and every a 0 , ..., a k ∈ R we have (2)    max 0 ≤ i ≤ k   i X n =0 a n g n   ∞ −   k X n =0 a n f l n   ∞    ≤ ε k X n =0 | a n | 2 n +1 . Below we gather the basic prop er ties of the notion of strong embedding. Prop ositi o n 6. L et X and Y b e c omp act metrizable sp ac es. Th e fol lowing hold. (i) If f ∈ SRC( X ) and g ∈ SRC( Y ) with g ≺ f , then g < f . (ii) If f ∈ SRC( X ) , g ∈ SR C( Y ) with g ≺ f and the se quenc e ( g n ) is a nor- malize d Schauder b asic se quenc e, then ther e exists L = { l 0 < l 1 < ... } ∈ [ N ] such that the se quenc e ( f l n ) is Schauder b asic and e quivalent t o ( g n ) . (iii) If f 1 ≺ f 2 and f 2 ≺ f 3 , then f 1 ≺ f 3 . (iv) If f ∈ SR C( X ) and g ∈ SR C( Y ) with g ≺ f , then φ Y ( g ) ≤ φ X ( f ) . (v) L et Z b e a c omp act metrizable sp ac e and e : Z → X ont o c ontinu ous. L et f = ( f n ) ∈ SRC( X ) and define, as in F act 2 , h = ( h n ) ∈ SRC( Z ) by h n ( z ) = f n ( e ( z )) for every n ∈ N and every z ∈ Z . If g ∈ SRC( Y ) is such that g ≺ f , then g ≺ h . Pr o of. (i) It is s traightforw ard. (ii) Let K ≥ 1 b e the basis constant of ( g n ). W e are go ing to show that there exists L = { l 0 < l 1 < ... } ∈ [ N ] such that ( g n ) is 2 K - e q uiv alent to ( f l n ). Indee d, let 0 < ε < 1 4 K and selec t L ε = { l 0 < l 1 < ... } ∈ [ N ] such that inequalit y (2) is satisfied. Let k ∈ N a nd a 0 , ..., a k ∈ R . Notice that (3)   k X n =0 a n g n   ∞ ≤ max 0 ≤ i ≤ k   i X n =0 a n g n   ∞ ≤ K   k X n =0 a n g n   ∞ . 8 P ANDELIS DODOS Moreov er, for every m ∈ { 0 , ..., k } w e have (4) | a m | ≤ 2 K   k X n =0 a n g n   ∞ as ( g n ) is a normalize d Schauder basic sequence (see [L T]). Plugg ing in inequalities (3) and (4) int o (2) w e get   k X n =0 a n f l n   ∞ ≤ K   k X n =0 a n g n   ∞ + 2 K ε   k X n =0 a n g n   ∞ ≤ 2 K   k X n =0 a n g n   ∞ by the choice of ε . Arguing similar ly , we see that 1 2 K   k X n =0 a n g n   ∞ ≤   k X n =0 a n f l n   ∞ . Thu s ( g n ) is 2 K -equiv alent to ( f l n ), as desir ed. (iii) It is a simple c a lculation, similar to that of par t (ii), a nd we prefer not to bo ther the r e a der with it. (iv) Let d ≥ 1. W e fix ε > 0 w ith ε < 1 2 d and w e selec t L ε = { l 0 < l 1 < ... } ∈ [ N ] such that inequality (2) is satisfied. F or every s = ( m 0 < ... < m k ) ∈ T d g we set t s = ( l m 0 < ... < l m k ) ∈ FIN( N ). Obs erve that for ev ery k ∈ N and ev ery a 0 , ..., a k ∈ R w e hav e 2 d k X n =0 | a n | ≥   k X n =0 a n f l m n   ∞ ≥ max 0 ≤ i ≤ k   i X n =0 a n g m n   ∞ − ε k X n =0 | a n | ≥   k X n =0 a n g m n   ∞ − ε k X n =0 | a n | ≥ 1 d k X n =0 | a n | − 1 2 d k X n =0 | a n | = 1 2 d k X n =0 | a n | . This y ields that t s ∈ T 2 d f . It follows that the map s 7→ t s is a monotone map from T d g to T 2 d f . Hence o ( T d g ) ≤ o ( T 2 d f ). As d w as arbitrary , this implies that φ Y ( g ) ≤ φ X ( f ), as desir ed. (v) It is also s traightforw ard, as the map e induces an isometric embedding of C ( X ) int o C ( Z ).  W e are going to present another prop erty of the notion of strong embedding which ha s a Ba nach space theor etic flav or. T o this end, we give the follo wing definition. A S TR ONG BOUNDEDNESS RESUL T 9 Definition 7. L et E b e a c omp act met r izable sp ac e and g = ( g n ) b e a b ounde d se qu enc e in C ( E ) . By X g we shal l denote the c ompletion of c 00 ( N ) under the n orm (5) k x k g = sup n   k X n =0 x ( n ) g n   ∞ : k ∈ N o . W e shall denote by ( e g n ) the standard Hamel basis of c 00 ( N ) r egarded as a se- quence in X g . Let us iso late some elementary prop erties of ( e g n ). (P1) The sequence ( e g n ) is a monotone basis of X g . Moreover, ( e g n ) is norma lized (resp ectively seminormalized) if and only if ( g n ) is. (P2) If ( g n ) is Schauder bas ic with bas is cons tant K , then ( e g n ) is K - e quiv alent to ( g n ). Less trivial is the fact (whic h we will see in the next section) that g ∈ SR C( E ) if and only if ( e g n ) is in SRC( K ), where K is the c losed unit ball o f X ∗ g with the weak* top ology . In light of prop erty (P2 ) a b ov e, the sequenc e ( e g n ) ca n be regarde d as a sort of “a pproximation” of ( g n ) by a Schauder bas ic sequence. The following pr op osition r elates the strong embedding of a s e q uence g = ( g n ) int o a s equence f = ( f n ) n with the e x istence of s ubsequences of ( f n ) which are “almost is o metric” to ( e g n ). Its pro of, whic h is left to the int erested r eader, is based on similar a r guments as the pro of of Pr op osition 6. Prop ositi o n 8. L et X and Y b e c omp act metrizable sp ac es, g = ( g n ) ∈ SRC( X ) and f = ( f n ) ∈ SRC( Y ) . If g str ongly emb e ds into f , then for every ε > 0 ther e exists L ε = { l 0 < l 1 < ... } ∈ [ N ] such that ( e g n ) is (1 + ε ) -e quivalent to ( f l n ) . 5. The main resul t W e ar e ready to state and prov e the strong bo undedness result for the class SRC. Theorem 9. L et A b e an analytic subset of SRC . Then ther e exists f ∈ SRC such that for every g ∈ A we have g ≺ f . W e r ecord the following consequence of Theorem 9 and P rop ositio n 8. Corollary 10. L et X b e a c omp act metrizable sp ac e and g = ( g n ) ∈ SRC( X ) . Then ( e g n ) is in SR C( K ) , wh er e K is the close d unit b al l of X ∗ g with the we ak* top olo gy. W e pr o ceed to the pro of o f Theo rem 9. Pr o of of The or em 9. W e fix a nor m dense sequence ( d n ) in the c lo sed unit ball of C (2 N ) such that d n 6 = d m if n 6 = m and d n 6 = 0 for every n ∈ N . W e also fix a sequence ( D n ) of infinite subsets of N such that D n ∩ D m = ∅ if n 6 = m and 10 P ANDELIS DODOS N = S n D n . Let A b e a n analytic subset of SRC and define ˜ A ⊆ N N by σ ∈ ˜ A ⇔ ∃ g = ( g n ) ∈ A ∃ ε > 0 such that  ∀ n ∀ k  k ∈ D n ⇒ k g n − d σ ( k ) k ∞ ≤ ε 2 k +1  and  ∀ n ∀ m  n 6 = m ⇒ σ ( n ) 6 = σ ( m )  . Then ˜ A is Σ 1 1 . Let T be the unique down w ards c losed, pruned tree on N × N such tha t ˜ A = pro j[ T ]. W e define a sequence ( h t ) t ∈ T in C (2 N ) as follows. If t = ( ∅ , ∅ ), then w e set h t = 0. If t ∈ T with t 6 = ( ∅ , ∅ ), then t = ( s, w ) with s = ( n 0 , ..., n m ) ∈ N < N . W e set h t = d n m . Clearly k h t k ∞ ≤ 1 for every t ∈ T . W e notice the following prop erties of the sequence ( h t ) t ∈ T . (P1) F or every σ ∈ [ T ] there exists g = ( g n ) ∈ A and ε > 0 such that for every n ∈ N and every k ≥ 1 with k − 1 ∈ D n we hav e k g n − h σ | k k ∞ ≤ ε 2 k . (P2) F or every g = ( g n ) ∈ A and every ε > 0 there exists σ ∈ [ T ] s uch that fo r every n ∈ N a nd every k ≥ 1 with k − 1 ∈ D n we hav e k g n − h σ | k k ∞ ≤ ε 2 k . W e pick an embedding φ : T → 2 < N such that for all t , t ′ ∈ T we ha ve φ ( t ) ⊏ φ ( t ′ ) if a nd only if t ⊏ t ′ . Let also e : T → N b e a bijection such that e ( t ) < e ( t ′ ) if t ⊏ t ′ for all t, t ′ ∈ T . W e en umerate the no des of T as ( t n ) acco rding to e . Now for every n ∈ N we define f n : 2 N × 2 N → R b y (6) f n ( σ 1 , σ 2 ) = χ V φ ( t n ) ( σ 1 ) · h t n ( σ 2 ) where V φ ( t n ) = { σ ∈ 2 N : φ ( t n ) ⊏ σ } . Clearly f n ∈ C (2 N × 2 N ) a nd k f n k ∞ ≤ 1 for all n ∈ N . Moreov er, it is eas y to chec k that f n 6 = f m if n 6 = m . It will b e co nv enien t to a dopt the following notation. F or every function g : 2 N → R a nd ev ery τ ∈ 2 N by g ∗ τ : 2 N × 2 N → R we shall denote the function defined by g ∗ τ ( σ 1 , σ 2 ) = δ τ ( σ 1 ) · g ( σ 2 ) for every ( σ 1 , σ 2 ) ∈ 2 N × 2 N ( δ τ stands for the Dirac function at τ ). Claim 11 . We have ( f n ) ∈ SRC (2 N × 2 N ) . Pr o of of Claim 11. B y the Main Theorem in [Ro2], it is eno ug h to show that every subsequence of ( f n ) has a further point wise co nv ergent subsequence. So, let N ∈ [ N ] ar bitrary . By Ramsey’s theorem, there exists M ∈ [ N ] such that the family { φ ( t n ) : n ∈ M } either consists of pa irwise incompar able no des, or of pairwis e compar able. In the first ca se we see that the sequence ( f n ) n ∈ M is po int wis e convergen t to 0. In the second case we notice that, by the pro pe rties of φ and the enumeration of T , for every n, m ∈ M with n < m we hav e t n ⊏ t m . It follows that there exists σ ∈ [ T ] such that t n ⊏ σ for every n ∈ M . W e may also assume that t n 6 = ( ∅ , ∅ ) for all n ∈ M . By prop er t y (P1 ) ab ove, there exist g = ( g n ) ∈ A , ε > 0 and a sequence ( k n ) n ∈ M in N (with p ossible rep etitions) such that k g k n − h t n k ∞ ≤ ε 2 | t n | for all n ∈ M . As g ∈ SRC, ther e exists L ∈ [ M ] such that the sequence ( g k n ) n ∈ L is p oint wise co nv ergent to a Ba ire-1 function g . By the fact that lim n ∈ L | t n | = ∞ , w e get that the sequence ( h t n ) n ∈ L is also po int wis e convergen t A S TR ONG BOUNDEDNESS RESUL T 11 to g . Finally notice that the s e quence ( χ V φ ( t n ) ) n ∈ L conv erges p oint wise to δ τ , wher e τ is the unique element of 2 N determined by the infinite chain { φ ( t n ) : n ∈ L } o f 2 < N . It follows that the sequence ( f n ) n ∈ L is p oint wise conv ergent to the function g ∗ τ . The claim is prov ed. ♦ Claim 12 . F or every g = ( g n ) ∈ A , g top olo gic al ly emb e ds into ( f n ) . Pr o of of Claim 12. Let g = ( g n ) ∈ A . By prop erty (P2) a bove, there exists σ ∈ [ T ] such that for every n ∈ N and every k ≥ 1 with k − 1 ∈ D n we ha ve k g n − h σ | k k ∞ ≤ 1 2 k . By the choice o f φ , we see tha t there e x ists a unique τ ∈ 2 N such that φ ( σ | k ) ⊏ τ for all k ∈ N . Fix n 0 ∈ N . By the fact that ther e e xist infinitely ma ny k with k g n 0 − h σ | k k ∞ ≤ 1 2 k , arguing a s in Claim 11 we get that the function g n 0 ∗ τ b elo ngs to the closure of { f n } in R 2 N × 2 N . It follows that the map { g n } p ∋ g 7→ g ∗ τ ∈ { f n } p is a homeomo r phic embedding and the claim is prov ed. ♦ Claim 13 . F or every g = ( g n ) ∈ A , g str ongly emb e ds into ( f n ) . Pr o of of Claim 13. Fix g = ( g n ) ∈ A . By Claim 12, it is enough to show that for every ε > 0 there exists L ε = { l 0 < l 1 < ... } ∈ [ N ] such that inequality (2) is satisfied for ( g n ) a nd ( f l n ). So, let ε > 0 arbitra ry . Inv oking pr op erty (P2) a b ov e, we se e that there exist σ ∈ [ T ] suc h that for every n ∈ N and every k ≥ 1 with k − 1 ∈ D n we hav e k g n − h σ | k k ∞ ≤ ε 2 k . There exists D = { m 0 < m 1 < ... } ∈ [ N ] with m 0 ≥ 1 and s uch tha t m n − 1 ∈ D n for every n ∈ N . By the prop erties of the enum eratio n e of T , there e xists L = { l 0 < l 1 < ... } ∈ [ N ] such that t l n = σ | m n for every n ∈ N . W e isolate, for future use, the following facts. (F1) F or every n ∈ N we hav e k g n − h t l n k ∞ ≤ ε 2 m n ≤ ε 2 n +1 . (F2) F or every n, m ∈ N with n < m we hav e t l n ⊏ t l m . W e claim that the sequences ( g n ) and ( f l n ) satisfy inequality (2) for the g iven ε > 0. Indeed, let k ∈ N a nd a 0 , ..., a k ∈ R . B y (F1) ab ove, for every i ∈ { 0 , ..., k } we hav e      i X n =0 a n g n   ∞ −   i X n =0 a n h t l n   ∞    ≤ ε i X n =0 | a n | 2 n +1 . This implies that    max 0 ≤ i ≤ k   i X n =0 a n g n   ∞ − max 0 ≤ i ≤ k   i X n =0 a n h t l n   ∞    ≤ ε k X n =0 | a n | 2 n +1 . The above ineq uality is a consequence of the following e le mentary fa ct. If ( r i ) k i =0 , ( θ i ) k i =0 and ( δ i ) k i =0 are finite seq ue nc e s of p o sitive reals such that | r i − θ i | ≤ δ i for all i ∈ { 0 , ..., k } , then   max 0 ≤ i ≤ k r i − max 0 ≤ i ≤ k θ i   ≤ max 0 ≤ i ≤ k δ i . 12 P ANDELIS DODOS So the claim will b e proved once we show that max 0 ≤ i ≤ k   i X n =0 a n h t l n   ∞ =   k X n =0 a n f l n   ∞ . T o this end w e a rgue as fo llows. F or every t ∈ T the function h t is contin uous. So there exist j ∈ { 0 , ..., k } and σ 2 ∈ 2 N such that max 0 ≤ i ≤ k   i X n =0 a n h t l n   ∞ =   j X n =0 a n h t l n ( σ 2 )   . By (F2), we hav e t l 0 ⊏ ... ⊏ t l k . Hence, by the proper ties of φ , we see that φ ( t l 0 ) ⊏ ... ⊏ φ ( t l k ). It follows that there exists σ 1 ∈ 2 N such that χ V φ ( t l n ) ( σ 1 ) = 1 if n ∈ { 0 , ..., j } while χ V φ ( t l n ) ( σ 1 ) = 0 otherwise. So   k X n =0 a n f l n   ∞ ≥   k X n =0 a n f l n ( σ 1 , σ 2 )   =   j X n =0 a n h t l n ( σ 2 )   . Conv ersely , let ( σ 3 , σ 4 ) ∈ 2 N × 2 N be such that   k X n =0 a n f l n   ∞ =   k X n =0 a n f l n ( σ 3 , σ 4 )   . W e notice that if χ V φ ( t l n ) ( σ 3 ) = 1 for so me n ∈ N , then for every m ∈ N with m ≤ n we a lso have that χ V φ ( t l m ) ( σ 3 ) = 1. Hence, there exists p ∈ { 0 , ..., k } such that χ V φ ( t l n ) ( σ 3 ) = 1 if n ∈ { 0 , ..., p } while χ V φ ( t l n ) ( σ 3 ) = 0 otherwise. This implies that   k X n =0 a n f l n   ∞ =   k X n =0 a n f l n ( σ 3 , σ 4 )   =   p X n =0 a n f l n ( σ 3 , σ 4 )   =   p X n =0 a n h t l n ( σ 4 )   ≤ max 0 ≤ i ≤ k   i X n =0 a n h t l n   ∞ and the claim is prov ed. ♦ As 2 N × 2 N is homeomor phic to 2 N , by Claims 1 1 and 13 and inv oking Prop ositio n 6(v), the pro o f of the theorem is completed.  References [AD] S. A. Argyros and P . Do dos, Genericit y and amalgamation of classes of Banach sp ac es , Adv. M ath., 209 (2007), 666-748. [ADK] S. A. Ar gyros, P . Do dos and V . Kanellop oulos, A classific ation of sep ar able R osenthal c omp acta and its applic ations , Dissertationes M ath., 449 (2008), 1-52. [A T] S. A. Argyros and S. T odorˇ cevi ´ c, Ra msey Metho ds in Analysis , Adv anced Courses in Math- ematics, CRM Bar celona, Birkh¨ auser, V erlag, Basel, 2005. [DF] P . Do dos and V. F erenczi, Some str ongly bo unde d classes of Banach sp ac es , F und. Math., 193 (2007), 171-179. [Ke] A. S. Kechris, Classic al D escriptive Set The ory , Grad. T exts in Math., 156, Springer-V erlag, 1995. A S TR ONG BOUNDEDNESS RESUL T 13 [KW] A. S. Kec hris and W. H. W o o din, A str ong b ounde deness the or em f or dilators , A nnals of Pure and Applied Logic, 52 (1991), 93-97. [L T] J. Lindenstrauss and L. Tzafriri , Classic al Banach sp ac es I and II , Springer, 1996. [Ro1] H. P . Rosentha l, A char acterization of Banach sp ac es not c ontaining ℓ 1 , Pr oc. Nat. Acad. Sci. (USA), 71 (1974), 2411-2413. [Ro2] H. P . Rosentha l, Pointwise c omp act subsets of t he first Bair e class , Amer. J. M ath., 99 (1977), 362-378. [T o] S. T o dorˇ cevi ´ c, T opics in T op olo gy , Lecture N otes in M ath., 1652, Springer-V erl ag, 1997. Universit ´ e Pierre et Marie Curie - P aris 6, Equipe d’ Ana lys e Fonctionnelle, Bo ˆ ıte 186, 4 place Ju ssieu, 75252 P aris Cedex 05, France. E-mail addr ess : pdodos@math. ntua.gr