A geometrical characterization of reflexivity in Banach spaces

Pub . Ma . UAB Vol . 30 ns 2-3 Des . 1986 a l e span (an-an+1)ncN A GEOMETRICAL CHARACTERIZATION OF REFLEXIVITY IN BANACH SPACES M .J .Chasco and Es eban Indu ain Summa y : Themain esul in his pape is he equi alence, o any Banachspace B , be ween (i) "E e y no malized basic sequence (an)nEN in B is weakly null" , and (ii) "Fo e e y no malized basic sequence (an)nFN in B , Pelczy iski p o ed ha (i) cha ac e izes he ac o B being e lexi e . So, he same holds o (ii) and we ha e a "geome ical" cha ac e iza ion o e lexi i y . We inish quo ing some equi alen e sion o he abo e esul . AMS Subjec Class . 1980 : 46 b 10 , 46 b 15 Keywo ds : Re lexi e Banach spaces, basic sequences, sequence o di e ences . 1 . P e ious Concep s . -Le B deno e a Banachspace and K i s scala ield, N, he se o na u al numbe s,  . .~ "closed linea span", and = (an)n -N be a linea ly independen sequence o ec o s in B . Call K( )  = n Can  ,  a n+1  ,  . . .I  ( ke nel o )  and neN K s ( ) = {K( ') ; ' is ke nel o ) is' no malized i 11 an 1 (n e N) is basic i he e is a unique sequence o scala s (Xn)neN such ha The sequence (an-an+1)neN is called . sequence o di e ences o I . is said obeweaklycon e gen o x e B i lim (a n ) = ( . x), o n e e y e B* (dual o B) . * is said obe minimal i he e exis s a sequence (a*) in [!~ wi h a* (a m ) = Ó(K onecke indices) , and uni o mlyminimal i i also e i ies  sup T an 11 . .11 a h 2 . The main esul . The esul leans on he ollowing wo lemmas Lemma 1 : E e y subsequence ' o a gi en sequence = (an)neN has ze o s ic ke nel i and only i he no malized sequence N =(an/IIanII)n has no subsequenceweakly con e gen o some ec o dis inc om ze o . P oo- : See ¡Ti , p . 172 . n an á o e e y x e subsequence (in ini e) o } (s ic Lemma 2 : Le = (an)neN be a minimalsequence wi h ze o ke nel . Le x e [J] such ha he se Sx = {kcN ; ak(x) ~0} is in ini e . We no e Sx  (pn ) ncN ' Then n x e K (( 1 a* (x) a )  )  i and only i he sequence s h=1 ph  Ph neN n a* (x) a )  is weakly con e gen ox h=1 Ph  PhneN P oo : (See II-TI) . I ollows om lemma 1 and he hi d F éche ' s axiom o con e gence (see IKI) . Now, we inally ha e he Theo em : Le B be a Banach space . Then he ollowing's a emen s a e equi alen (i)  B is e lexi e , (ii) E e y no malized basic sequence (an)neN in B is weakly con e gen o ze o , (iii) E e y no malized basic sequence (a n ) nEN in B e i ies a l e an-an+1 ; n e NI P oo , : In IPI has been p o ed ha (i) is equi alen o (ii) . -(ii) implies (iii) is ob ious , conside ing n-1 a l - a n =  (ai-ai+1) i=1 (iii) implies (ii) : -Suppose ha o e e y no malized basic sequence = (an)ncN a l e [an -a n+1 ; n E Ni . No ice ha a l e Can-an+1 ; n e N] i and only i a l e K((al-an)n) (see, o ins ance, IR¡, p oposi ion 2 .2) Take (pn ) nF_N a subsequence o N , wi h p1 = 1 . By hypo hesis, he sequence (apn)neNalso e i ies a l s [a pn - a pn+l ; n e N1 , so , i ollows ha a l e K5((al-an)nEN) Now, applying lemma 2 o a l and (a n -a n+1 ) neN ' we_ha e ha (al-an)nFN is weakly con e gen o a l , and he e o e (an)ncN is weaklycon e gen o ze o . 3 . Equi alen e sions . -In ICH-II (p ep in o his pape ) he ollowingequi alen e sions o he heo em a e gi en (i )  Can  ;  n e Ni = Ca n -a n+l  ;  n e N ]  ,  o e e y no malized basic sequence (an)neN in B , ( )  Le (an)nEN be a no malized basic sequence in B . Then , i s sequence o di e ences canno be uni o mlyminimal , ( i) Po e e y no malized basic sequence (an)ncN in B , Ca n-an+1  ;  n e N J  canno be an hype plane in Can  ;  n e N~ Acknowledgemen . We hank he e e ee o his aluable sugges ions . 4 . Re e ences . ICH-II CHASCO, M .J . - INDURAIN, E .  : Ca ac e izacionesgeomé icas de la e lexi i- Ma . Uni . Pa ia 4(4) 165-204 (1978) . Rebu el 18 d'Agoe de 1986 Depa amen o de Geome ía y Topología Facul ad de Ciencias 50009-ZARAGOZA ESPAÑA dad en espacios de Banach . (P ep in ) . Pub . S . Ma . Ga cíade Galdeano . Se ie II . Sección 1 n9 102 . Za agoza 1986 II-TI INDURAIN, E . - TERENZI, P . : A cha ac e iza ion o basic sequences in Banach spaces . Rend . Ace . Naz . dei XL 1049 ol X 1986 ( o appea ) IKI KURATOWSKI, K . : Topologie, ol I . PWN Wa sam 1952 P PEZCZY VSKI, A . : A no e on he pape o I . Singe "Basic sequen- ces and e lexi i y o Banach spaces" . S udiaMa h . 21, 371-374 (1962) . IRI REYES, A . : A geome ical cha ac e iza ion o Schaude basis . A ch . Ma h . 39 , 176-179 (1982) . ¡Ti TERENZI, P .: Bio hogonalsys ems in Banachspaces . Ri .