Iterated series and the Hellinger-Toeplitz theorem

Publicacions Ma emá iques, Vol 36 (1992), 167-173 . A bs ac A p oblem which is equen ly encoun e ed in analysis is he in e - changing o he summa ions in an i e a ed double se ies . i aij E IR, i, j E PN is a double sequence, when does he equali y ij hold? Fo example, i a ij > 0 o all i, j, hen his condi ion holds (whe e he sums may be in ini e), and, in gene al, i i=1j=1 j=1i=1 00 00 ITERATED SERIES AND THE HELLINGER-TOEPLITZ THEOREM CHARLES SWARTZ We show ha an i e a ed double se ies condi ion due o An osik implies he uni o m con e gen e o he double se ies . An applica- ion o An osik's condi ion is gi en e he de i a ion o a ec o o m o he 1 - lellinge -Toepli z Theo em . Tha is, 57 1 : l aij 1 < oo, hen he equali y holds ([4]) . Ano he condi ion which i-1 j-1 gua an ees he equali y is he exis en e o he double limi , lim 00 00 along wi h he con e gen e o he se ies Eaij, E aij ([4]) . The exis- j=1 i-1 en e o his double limi is o en di iclu o e i y ; one possible way o gua an ee he exis en e o he double limi is o show ha one o he i e a ed se ies is uni o mly con e gen , bu his is also o en di icul o e i y . In his no e we would like o poin ou he exis en e o a condi- ion due o P . An osik which in ol es only he i e a ed se ies and which gua an ees he exis en e o he double limi and, hence, he equali y o he wo i e a ed se ies ([1]) . An osik's condi ion wo ks equally well o ec o - alued se ies so we p esen his e sion . To illus a e he u ili y o An osik's condi ion, we es ablish a Hellinge -Toepli z ype heo em 168  CH . SWARTZ conce ning he con inui y o ma ix ans o ma ions be ween sequence spaces . Th oughou his no e G will deno e an Abelian opological g oup . Le xij E G o i, j EN . We assume ha he se ies > J xij (> J xij) con e ges o each i E Nl (j E IN) and seek candi ions which gua an ee he equali y 00 00  w 00 (and exis en e) o he wo i e a ed se ies > , > , xij, L > , xij .  One j=1 m n such condi ion is he exis en e o he double limi li n ~- mn u , i=1 j=1 x ij and called he double se ies gene a ed by xij) ([4]) . We gi e ij a condi ion which in ol es only i e a ed se ies and which gua an ees he exis en e o he double se ies . Since his condi ion in ol es only he i e a ed se ies, i may some imes be easie o check han he exis en e o he double limi . We gi e an example o such a si ua ion in p o ing he Hellinge -Toepli z esul gi en in Theo em 3 . Recall ha a se ies ~ xn con e ges in G o e e y subsequence {ni} . The p incipal ool i=1 used in he p oo o ou main esul on double se ies is he ec o - alued gene aliza ion o subse ies con e gen o he classical Schu Lemma en weakly con e gen se ies in 1 1 ([2, 8 .1]) . xi,, con e ges o - cae[¿ inc easing se- J=I . quence o posi i e in ege s {m j } .  Then he double se ies Theo em 1 . Suppose e ges an,d 00 00 di e ence be ween he wo se ies x i in G is subse ies con e gen i he se ies x ij = LL xii . i=1 j=1  j=1 i=1 P oo - No e ha he se ies  xij con e ges o each j (conside he i=1 ce 00 xi,,, whe e i=1 j=1  i=1 j=1 nk = k 'o each kand {mk} is he subsequence {1, . . . , j - 1, j + 1 . . . . }) . xin~ an 00 00 j (deno ed con- IT RAT n SERIES  169 00 00 Thus, i a C IN is ini e, he meaning o i=1 jEo in ini e, a ange he elemen s o a in o a subsequence {ni, n2, . . . } and se 00 00  00 0o  m xij = 1 : Y :xinj . Se z n ,, j = 57 xij . Then o a C N, Y : Zncj = m 8 .1]), he se ies jEo i=1 exis s and equals ij con e ges o =1jE o i=1  jEo as m -> oo . By Schu 's Lenl la ([2, ;) is subse ies con e gen and li n m 00 00  00 00 xij is clea , i a C N! is j=1 i=1  i=1 jEo n , n xij uni o mly 'o a C IN . Hence, he double li ni liin > , > , X¡33 m ' ¿ Example 2 . The condi ion in Theo em 1 is su icien o he exis ence o he double ; se ies (and he equali y o he 2 i e a ed se ies), bu ¡ is no necessa y . In he scala case he hypo hesis o Theo e l 1 iinplies ha he ows o he ma ix [x ij ] a e absolu ely con e gen , so he ma ix xij = (-1)j+ l /i 2 j a¡ls o sa is y he lypo hesis o Tlleo cin 1, bu i e ; double se ies 5-- x ij con e ges . Z I .7 Theo em 1 was p o ee o se ies in a space ; cquippcd wi h a scqucn- ial con e gence s uc u e sa is ying ce ain con e gence p ope ies by An osik in [1, 3 .3] . An in e es ing aspec o he hec em, e en o scala, - alued se ies, is ha he condi ion in he hypo hesis o he heo e l o lly in ol es i e a ed se ies and, he e o e, can some imes be easily checked . We gi e an example o such a condi ion in he p oo o he Helli lge- ; - Toepli z esul below . The classical Helli lge -Toepli z Theo e l asse s lla a iy ma ix which maps 12 ¡ i o 12 is (no m) con inuous ([5]) ; wc seek condi ions on sequence spaces which will gua an ee ha ma ix ans 'o ml io ls be ween he sequence spaces a e con inuous . Since Theo em 1 is alid o ec o - alued se ies, we conside ec o - alued sequence spaces . Le X, Y be Hausdo opological ec o spaces and le L(X ; Y) be he space o all con inuous linea ope a o s om X in o Y . Le E(F) be a ec o space o X- alued (Y- alued) sequences ; i x E E, we deno e ; he k ` h coo dina e o x by xk so x = {xk} . The,-dual o E (wi h espec o Y), deno ed by E", is he space o all sequences {Tk} =TC L(X, Y) such 170  C11 . SWARTZ ha he se ies k=1 equi e ha he ope a o s a e con inuous) ; i Y is lle scala ield, we 00 w i e E l 9y = Ep . I x E E and T E Ep y , we w i e T - x = Tkxk . k=1 This gi es a map x - T - x om E in o Y, and we le u (E, EOY) be he weakes opology on E such ha all o heses maps o T E EQY a e con inuous ; when X and Y a e he scala ield, his is jus he weak opology 0'(E, EA) om he duali y be ween E and i s /O-dual, Ep . Le Aij E L(X, Y) o i, j E N and le A be he ope a o - alued ma ix [Aij] . We say ha A maps E in o F o A E (E, F) i o each x E E, i E 00 Tkxk con e ges o each .x E E ([9], Maddox does no IN, he se ies E Aijxj con e ges and he sequence {E Aijxj } E F, Le ., j=l  j=1 i he o mal ma ix p oduc Ax= { j=1 xE E . We a e in e es ed in condi ions which gua an ee ha a ma ix A E (E, F) is con inuous wi h espec o app opia e opologies on E and F . Fo example, he classical Hellinge -Toepli z Theo em asse s ha any (scala ) ma ix A E (l2, 12) is no m con inuous ([5]) . Toepli z and Kó he gene alized his esul o o he sequence spaces ([8], [7, 34 .7(7)]) . We now use Theo em 1 o gi e a u he gene aliza ion o he Toepli z- Kó he esul o ec o - alued sequence spaces . The pai (X, Y) is said o ha e he Banach-S einhaus p ope y i when- e e Tk E L(X, Y) con e ges poin wise, lim Tkx = Tx, o x E X, hen he limi ope a o T is con inuous . Fo example, i X is an F-space o i X is a ba elled locally con ex space and Y is a locally con ex space, (X, Y) has he Banach-S einhaus p ope y . The sequence space E is said o be mono one i moE = E, whe e mo is he scala sequence space consis ing o all sequences wi h ini e ange and moE is he coo dina ewise p oduc o sequences in mo and sequences in E ([3]) . In pa icula , any no mal (scala ) sequence space is mono one ([6, 30 .1]) . Fu he , coo (X) deno es he space o X- alued sequences which a e 0 e en ually ; i X is he scala ield we w i e coo(X) = coo . We now es ablish ou Hellinge -Toepli z esul , which asse s ha a ma ix ans o ma ion A E (E, F) is con inuous wi h espec o he weak opologies o,(E, E AY ), Q(F, FO Y ) o E and F unde app op ia e condi ions on E and F . Aijxj} belongs o F o each Theo em 3 . Le E he mono one and con ain coo(X) and le (X, Y) ha e he Banach-S einhaus P ope y . I he ma iz A = [Aij] maps E in o F, hen A is u(E, EO Y ) - o,(F, FO Y ) con inuous . P oo . . Le B= {Bi } E FP Y and le Ai be he ieh ow o A so B . Ax = Bi(A' - x) _ BiAijxj o x E E . No e o each j he se ies j BiAij con e ges in he s ong ope a o opology o L(X, Y) (Fix j and o x E X le x be he ec o in E wi h x in he jeh coo dína e and 0 elsewhe e .  Then i : Aikxk = Aijx so {A ij x}i E F and since k A ij x con e ges, Le ., E BiAij con e ges in he s ong i ope a o opology o an elemen o L(X, Y) since (X, Y) has he Banach- S einhaus P ope y) . Since E is mono one, he se ies ~ J ~ B i A j ,~, xn , j BEFQ Y , ITERATED SERIES  171 con e ges o each xE E and subsequence {n j } .  By he In e change Theo em 1, i we se Cj = i : BiAi ; and C = {C j }, hen C E EQ Y and B-Ax=i : i j  j i E which is a (E, EQ Y ) con e gen o 0, hen Ax óis u (F, F 13 Y) con e gen o 0 and A is con inuous wi h espec o hese opologies . B i A ij x j = EE B i A ij x j = C - x so i {x 6 } is a ne in The (scala ) space 12 ob iously sa is ies he hypo hesis o Theo em 3 and i A E (12,12), hen A is weakly con inuous and, hence, no m con inuous ; his is jus he classical Hellinge -Toepli z Theo em . Mo e gene ally, i E and F a e scala sequences and E is mono one and con~ ains coo, hen any ma iz map A : E ---> F is con inuous wi h espec o he weak opologies o,(E, EQ) and o,(F, Fa) . In pa icula , i E is no mal (Le ., i x E E and Jyk1 < jxk1 o all k, hen y E E), hen E is ob iously mono one so he esul holds in his case ; his is essen ially he e sion o he Hellinge -Toepli z Theo em o sequence spaces due o KS he ([7, 34 .7 .(7)]) . Ko he's esul uses a-duals ( he o¿-dual o a scala sequence space E is he se o all sequences {yj } such ha _ jxj yj 1 < oo o all x E E ; o mono one spaces E° = EA) so is so newha weake han Theo em 3 since u(F, Fp)'is s onge han u(F, Fa) . In he scala case he p oo o Theo em 1 abo e uses only he scala e sion o he Schu Lemma ([2, 8 .2]) and he p oo o Theo em 3 is hen much mo e elemen a y han ha gi en by KS he which uses esul s on p ojec i e limi opologies o locally con ex spaces . A scala e sion o Theo em 3 has been es ablished in [11] . I Y is he scala ield, he spaces E and EAY =E 13 a e in duali y wi h 172  Ci-1 . SWAlUZ espec o he bilinea pai ing x - y, whe e x e E and y is an X'- alued sequence belonging o EQ ([10]) . I E is mono one and X is ba elled, he hypo hesis o Theo em 3 a e sa is ied so any ma ix map A om E in o F is con inuous wi h espec o he weak opologies u(E, EO) and u(F, FO) . In his case, which includes he case when X is also he scala ield, he ma ix mapA o E in o F is also con inuous wi h espec o he Mackey (s ong) opologies o E and F, espec i ely ([10]) . Mo eo e , he co npu a ion in Theo em 3 shows ha he anspose o he ope a o A, A' : FQ -> EA, is gi en by he ma ix [A íi ] . In his case, he anspose map, A', is con inuous wi h espec o he weak (Mackey, s ong) opologies o FQ and El 3 , espec i ely ([7, 32 .2]) . The e a e abundan examples o ec o sequence spaces sa is ying he hypo hesis o Theo em 3 . Fo example, coo(X) o co(X), he ec o space o all X- alued sequences which con e ge o 0, o mo(X), he ec o space o a,ll X- alued sequences wi h ini o ; ange, o l`w(X), he space o all X'- alued bounded sequences, a e all mono one sequence : spaces con aining coo(X ) . I X is a no med space a d 1 < p < oc, he space mono one and con ains coo(X) . Re e ences lp(X) consis ing o all X- alued sequences such ha E 11xkl1 p< oo is k- 1 .  P . ANTOSIK, "On in e change o limi s, gene alized ,nc ions, con- e yen,ce s uc u es and hei applica ions," Plenum P ess, N .Y ., 1988, pp . 367-374 . 2 . P . ANTOSIK AND C . SWARTZ, "Ma iz me hods in Analysis," Sp inge Lec u e No es in Ma liema ics 1113, Sp inge -Ve lag, Hei- delbe g, 1985 . 3 .  G . BENNI-TT, A new class o sequence spaces wi h applica ions in sumniabili y l co y, J . eine angew . Ma h . 266 (1974), 49-75 . 4 .  T .J . Bizomwicn, "An in oduc ion ,o he heo y o, in ni e se ies," MacMillan, London, 1926 . 5 .  E . H a .i,INCC : AND O . TO p .<< z, G ündlagen ü Bine Theo ie den une dlichen Ma i en, Ma h . Ann . 69 (1910), 289-330 . 6 .  G . Kó" uE, "Topological ec o spaces I," Sp inge -Ve lag, N .Y ., 1969 . 7 .  G . KÓTIm, "Topological ec o spaces II," Sp inge -Ve lag, N .Y ., 1979 . IT aZATED SERIES  173 8 .  G . KóTiIG AND O . TOEPLITZ, Linea e Rliu ne mi unendlichen io- len Koo dina en und Ringe unendlichen Ma izen, J . eine angew . Ma h . 17 (1934), 193-226 . 9 .  I . MADDOX, "In ini o ma ices o ope a o s," Lec u e No es in Ma hema ics 786, Sp inge -Ve lag, Be lin, 1980 . 10 .  N . PHOUNG-CÁC, Su les espaces pa ai de sui es géné alises, Ma h . Ann . 171 (1967), 131-143 . 11 . C . SWARTz, Weak sequen ial comple eness o sequence spaces, p ep in . Depa me i o Ma hema ical Sciences College o A s and Sciences Box 30001 Dep . 3MB Las C uces NLW MGXICO 88003-0001 P ime a e s ió ebuda el 8 d'Ab il (le 1991, da7 e a e sió ebuda el 2 de Se eu b e de 1991