Infratopological bornologies and Mackey-convergence of series

Pub . Ma . UAB Vol . 30 n2 2-3 Des . 1986 INFRATOPOLOGICAL BORNOLOGIES AND MACKEY-CONVERGENCE OF SERIES MiguelA . Canela and Me cé Se ahima SYNOPSIS : In his pape some aspec s o ín a opological bo nologies a e discussed . Fi s , i is shown ha bo nologies wi h coun able basi's a e in a opological . Second, i is shown ha he con e gence o se ies p esen s some : pa hologies beyond his class . We , bllow closely he , e minology o [51 and [4] , whe e he main opics conce ning bo nological spaces can be , ound . Roughly speaking ;, a bo nology is iü a opological when :can be de ined om a opologyin he usual way : he bounded se s : a e hose which a e abso bed by ze o neighbou hoods . Mo e p ecisely  le (E, S ) be a linea bo nological space, and TS he Mackey ; closu e opolo- gy associá ed o his space . . The Von Neumann bo nology o TS , de- no ed by , BIT S , is de ined as ollows : a subse B C E' is bounded when o e e y ze o neighbou hood V, he e is some A >-0 such ha B C  X V  (T ; is abso bed by V)' . The bo nology 1 is said o be in- a opolog'cal when  0= BT0,  _ A mo e use ul cha ac e iza ion, whichdoessno in ol e any opology is he ollowing~ 12-1 i P oposi iom 1 . . A linea bo nological space ( E, 0  )  is - in a opo- logical i 'and only i e e y subse B C E, which is abso bed o e e y bo ni .Vo ous subse o & . is bounded in  (E,R) ( a subse UC E  is bo ni o ous when U abso bs e e y bounded se o (E, S )) . The boundedsubse s o an o dina y opological ec o space ( he Von Neumann bo nology) o m an in a opological bo nology . Ne e heless,bo nologies which a e no in a opological can appea when we deal wi h equicon inuous se s . We e e o (1 ; _ . , example p . 166, o an equicon inuous bo nology no Kolmogo o and consequen ly no in a opological . We see nex ha spaces wi h coun able basis a a also in a opo- logical . P oposi ion 2 . _Le _ (E, S ) be a sepa a ed con ex bo nological space and suppose ha S admi s a coun able basis  (Bn ) n > 1 . Then Bis in- a opological . P oo . We can supposse ha he B n 's a e absolu ely con ex and ha he sequence is inc easing . Thus, i we deno e by ~~ . ~~ n he gauge o B n , we ob ainan inc easingsequenceo no ms, each ~~ . lin de ined in E n = Span B n .To make i easie , we de ine llxli n o x E E En . Le A be an unbounded subse o (E, S ) . hen, o e e y n, we can ind xn E A wi h llxn 11 n > n2 . Fo e e y k, we de ine : and pu ing : ñk = 1 min (lixlll k ', . . ., llxkllk ' l) . k V = U  ñ k . B k , k 11 we ob ain a bo ni o ous subse o E, and V does no abso b A, be- cause, o E> 0 a bi a y,i n > E :-l, xn does no belong o any AkBk : a)  i k 2 n, we ha e  11 E xnllk 1 Ek ñ k a E n A k > Xk b)  i  1  _ k  <  n,  we  ha e  11  Ex n1 ik ?  11  Ex n l I n > en 2  >  n >X k* ApplyingP oposi ion 1, we conclude ha (E, S ) is in a opo- logical . People which is used o linea bo nological spaces knows ha such a space can ail o ha e he p ope ies ha one hopes o ind in Func ional Analysis, unless some ex a assump ions a é aken . Res ic ion o in a opological spaces is en example o ex a assump- ion unde which some pa ologies do no appea . We will see he e ha he con e gence o se ies does no wo k in he usual way when we pla- ce ou sel es ou side he class o in a opological bo nologies . A se ies  2 :  x nis said o be Mackey-con e gen in a linea n > 1 bo nological space when he sequence , ( Y-  x k) n> 1 o pa ial k=1 sums is Mackey-con e gen ."The ollowing esul s a e well-known o he opological con e gence in a opological ec o space . Ou con- e gence has been conside ed in [6] . P oposi ion 3 . ( i ) Le (E, S ) be a Mackey-comple e con exbo - nological space, (xn ) n Z 1 a bounded sequence in (E B ), and (an ) n  > 1  a sequence o eal numbe s such ha  lan 1  is i- n >_ 1 ni e . Then,  2 : an xn is Mackey-con e gen . n i 1 (ii) Le (E, S ) be an in a opological con ex - bo nological space, and (xn ) n > 1 ' a sequencesuch ha , o e e y sequence o eal numbe s (a n ) n >  1 wi h  la n 1  ini e,  he se ies  ,~  a n  .  x n  is Mac- n > 1  - n > 1 key-con e gen . Then (xn)n > 1is bounded . P oo . (i) . Take a bounded disk B con aining (x n ) n > l, and deno e is a Cauchy sequence wi h espec o 11 . II B , and, (E, S ) being Mackey-comple e, i is Mackey-con e gen in (E, B ) . (ü) I (x n ) n >_ 1  is unbounded_ he e is a bo ni o ous subse  V which does no abso b his sequence . Replacing (xn ) n  by a sub- á 1 sequencei necessa y n i s gauge . The sequenceo pa ial sums  n a k x k ) , 'k-1  n > 1 we can suppose ha x 11 EZ n 2 V o e e y n . Then (n ~2 . x n )  is no Mackey-con e gen o ze o . // na 1 As we ha e p e iously announced,be will show, h ough a coun- e example, ha he assump ionon (E, S ) in pa (ii) o he p e- ceding P oposi ion is no supe luous . Example 4 . Le E be he space o measu ab'le eal unc ions on he uni in e al I = - 10,1 (wi h he s anda d iden i ica ion), p o ided wi h he o de bo nology : a subse A C E is bounded i he e is some gE :E, wi h g a 0 and I l < g o e e y E A . This bo nology is con ex, bu no he associa ed in a opological bo nology . So E is no in a opological . De ail :s on his ac can be ' ound in [3J .  Mo eo e he Mackey-con e gence ela i e o he o de bo nology coincides wi h he almos e e ywhe e con e gence . We conside he sequence ( n )  de ined as ollows : (we n ? 1 deno eby X T he cha ac e is ic unc iono a subse T o I) : 4 = 3 . X  5 = 3 . X  }  1  , and so on . 10,1/41  '  ( 4 . z- ho Ob iously, ( ) n  is bouhded in he o de bo nology . Ne e heless, n ? 1 1  an n con e ges almos e e ywhe ewhen  2 :  lan 1  con e ges . l su ices o conside he case in which a n 1 0 o all n . Mo eo e , i su ices, in' his case, o p o e ha  a n n con e ges in n '= 1 measu e, because o an inc easing sequence, bo h ypes o con e - gence a e equi alen . Fo each n, we deno e : Take a ixed n . On each in e al b n =  aj  gn =  aj j 2n-1 é j <2 n  2n-1 i j< 2n he cons an alue na j o some j, 2 n-1 5 j < 2n . The e o e, gn exceeds bn on his in e al i and only i a j >  n 1 b n . Bu his happens o a mos n-1 o hese in e als . Thus : 2 n-1 Take now an a bi a y E > 0, and choose N such ha : b n <  n-1  < E . 2 n 2 N  n2N k  k+11  he unc ion g akes 2n ' 2n  n Then : m( { x :  1  9n (X) > E })  <  m( {  x :g n (x)  >  b n })_`Z  n  <  E . n Z N  n > N  Zn-1 n =N This a gumen p o es ha  gn con e ges in measu e, and so n? 1 does  aj j .// j 21 R EFERENC E S 1 . M . A . Canela . Bo nologí a equicon inua en un espacio de aplicaciones lineales con inuas . Re is a de la Uni e sidad de San ande , núm .2, pa e 1 . (1979) (111-121) . 2 . M . A . Canela . Linea bo nologies and associa ed opologies . Collec . Ma h . 32 (1981), 165-178 . 3 . M . A . Canela . Som e classes o linea bo nologlcal spaces . oc . Royal Soc . Edinb . 90 (1981), 155-161 . 4 . í . Hogbe-Nlend . Théo ie des bo nologies e applica ions . Lec u e No es in Ma hema ics 213 (Be lin : Sp inge , 1971) . 5 . H . Hogbe-Nlend . Bo nology and unc ional analysis (Ams e dam : No h Holland, 1978) . 6 . V . B . Mosca elli . Bases in bo nologícal spaces . S udiaMa h . 50 (1974), 251-264 . Rebu el día 27 de . juny de 1986 Uni e si a de Ba celona Dep . d e TeO ia de Funcions Facul a de Ma ema iques G an Via, 585 08007-BARCELONA ESPANYA