ON BOUNDED SEQUENCES AND SERIES CONVERGENCE IN BANACH ABSTRACT SPACE

Transcription

1 Lic Paper. 69 Serie Metodi qatitativi dicembre 999 ON BOUNDED SEQUENCES AND SERIES CONVERGENCE IN BANACH ABSTRACT SPACE Roberto D Agiò. Itrodctio. Theorem 3 give two proof of a fficiet coditio for eqece covergece ad erie mmability i ormed or Baach abtract pace. Sch a coditio amot to a otio of eqece bodede called d-bodede (deep bodede which i deeper tha ordiary bodede. I fact firt ay d-boded eqece i boded wherea the covere doe ot hold (Theorem. Secod at variace with ordiary bodede d-bodede tr ot to be more retrictive tha covergece itelf. I fact d-bodede implie covergece wherea coverely if ad oly if a particlar coditio i met the a coverget eqece i d-boded (Theorem 4. I detail Theorem 3 verio (A prove that a fficiet coditio for eqece covergece ad erie mmability i ormed or Baach abtract pace i give by the followig ieqalitie which defie eqece d-bodede ( where ( ( D = cot. > 0 D ( D = + ( = i a eqece i ormed or Baach abtract pace ad the qatitie + are defied by ch a eqece a follow ( ( ( 0 ( (2 + ( ( + = ( The theorem metioed i the above are iter alia coeqece of a imple repreetatio relt (Lemma accordig to which ay eqece = defied i a liear pace S ca be repreeted a the eqece of the CH- mea (i.e. Ceàro-Hölder firt arithmetic mea ee e.g. [Ce] [Toe] [Zyg] [Wid pp. 3-2] [Str pp ]

2 Lic Paper. 69 dicembre 999 (3 of a related ad iqe eqece eqece ( ν = ν= = which we call the derlyig eqece of the give. The by (2-(3 d-bodede ( actally amot to the iform bodede of the followig ifiite-dimeioal matrix (4 ( ( = ( 2 ( 3 M ( ( 2 ( = ( 2 ( + ( 3 ( + M ( ( 3 ( 2+ ( 2 ( 2+ ( 3 ( 2+ M + = L L L where (I the firt colm i the give a eqece firt row i it derlyig eqece by (3 the movig average (5 ( ( = + ( ν ν= + the ettig = i (3 ad = (6 or by Lemma eqece (3 (II the ad (III the remaiig elemet of (4 are (2 or + = 2 ; i (5 we get bac the derlyig eqece ( ( = ( ( + = = 2 + i.e. the firt row of matrix (4; frther by (3 the elemet of the firt colm are CH- mea ad th give rie to the correpodig Toeplitz matrix [Toe] [Zyg] [Wid]. Moreover it tr ot (Theorem 2 that the d-bodede ( of a eqece of matrix (4 i eqivalet to (7 D = i.e. it i eqivalet to the ordiary bodede (7 of it derlyig eqece or the iform bodede. Th by Lemma ad Theorem the cla of the d-boded eqece i ot empty ad preciely it member are all the eqece (3 of CH- mea of all the boded eqece. I fact by Lemma ad Theorem Theorem 3 (ee verio (A above ha two frther eqivalet ad meaigfl verio (B-(C. Ideed o the oe had (B Theorem 3 prove that if the firt row of matrix (4 i.e. ay (poibly diverget eqece i boded the ch a eqece i CH- coverget (i.e. it coverge i the ee of Ceàro-Hölder amely the eqece (3 of the 2

3 Roberto D'Agiò O boded eqece ad erie covergece i Baach abtract pace CH- mea of (firt colm coverge i the ordiary ee. O the other had (C Theorem 3 prove that the firt colm of matrix (4 i.e. ay give eqece i the ordiary ee if the firt row i.e. the derlyig eqece coverge i boded i the ordiary ee. Th by a approach actally deiged for diverget eqece (or erie i.e. the Ceàro-Hölder method Theorem 3 cceed i provig a fficiet coditio for the ordiary covergece of eqece (or mmability of erie throgh the ordiary bodede of the derlyig eqece. Frthermore a ca be eaily ee verio (B agree with a claical relt of covergece ad mmability theory ee e.g. [o pp. 73-4] [Zyg p. 74] [Toe] [Meg pp ] [Woj p. 57]. However a frther iteretig featre of the above covergece ad mmability relt i that Cachy coditio Baach completee ad the Ceàro-Hölder mea (3 excepted their proof do ot ivolve ay deep relt or cotrct of the exitig covergece ad mmability theory. 2. A type of eqece bodede i Baach abtract pace. The itrodctory remar of Sectio motivate the followig DEFINITION. ( Deeply boded eqece i ormed abtract pace. Let ( S be a ormed pace. The a eqece ( hold = i aid to be d-boded (deeply boded if A a cadidate for a fficiet coditio of covergece d-bodede mt exactly behave a proved by Theorem below that i: wheever a eqece i d-boded the it i boded i the ordiary ee wherea the covere doe ot hold (a ecod proof of thi fact will be give a a coeqece of Theorem 3-4. THEOREM. Coider the abtract ormed pace ( S. The for ay eqece = (i d-bodede implie bodede (ii bodede doe ot imply d-bodede ad i particlar we have (8 ( ( 2D D + =. + 3

4 Lic Paper. 69 dicembre 999 Firt proof of Theorem. (i: by ( firt ieqality i particlar. (ii: by (2 ad a property of the orm we obtai (9 ( + ( + / ( + ( + th by the bodede of the eqece ; i.e. the firt ieqality i ( ad imple algebra (9 give (8 that doe ot agree with the ecod ieqality i ( Theorem 2-4 ret o the imple repreetatio Lemma that we prove below. It tate that ay arbitrary eqece = defied i a liear pace S ca be repreeted a the eqece of the CH- mea (Ceàro-Hölder firt arithmetic mea ee e.g. [Ce] [Zyg] [Wid pp. 3-2] [Str pp ] of a related eqece that we call the derlyig eqece of the give eqece (of core i immaterial whether i tr the derlyig eqece i ee a a eqece of partial m of a erie or ot. LEMMA. (Repreetatio of eqece i liear abtract pace by CH- mea. Coider the abtract liear pace S ad =. The ay eqece ca be repreeted i the form (3 i.e. a the eqece of the CH- mea of ome ad iqe eqece (which i called the derlyig eqece of the give eqece. The frther (5-(6 hold. Proof. Give a eqece (0 ( ( = we get a iqe eqece by ettig = = 3 Frthermore eqece (0 give bac iqely the give eqece repreeted i the form of the eqece of the CH- mea of eqece (0 itelf i.e. i the form (3. Ideed by obvio iterated iverio ad btittio (0 give iqely ( ( = ( = + / ( ν = ν= / which i the give eqece = i (3 ad = i (5 we get (6. = 3 i the form (3. The by (2-(3 (5 hold ad ettig Theorem 2 below give a eceary ad fficiet coditio for the d-bodede of a eqece throgh the ordiary bodede of it derlyig eqece. I fact Theorem 2 prove that the d-bodede property ( of a eqece bodede (7 of it derlyig eqece i eqivalet to the ordiary. Th by Lemma ad Theorem the 4

5 Roberto D'Agiò O boded eqece ad erie covergece i Baach abtract pace cla of the d-boded eqece i ot empty ad preciely it member are all the eqece (3 of CH- mea of all the boded eqece. THEOREM 2. (Neceary ad fficiet coditio for eqece d-bodede i ormed abtract pace. Coider the ormed abtract pace ( S. Let be the derlyig eqece of a eqece atifie (. ad =. If ad oly if atifie (7 the Proof. Neceity. By (6 of Lemma if ( hold the we get (7. Sfficiecy. By (3 ad (5 of Lemma ad the triaglar property of the orm if (7 hold the we get (; ideed ( ν = ( ν / ν = ν = ( ( + = ( ν + ( ν / / D + / ν= + ν= + D 3. Covergece by eqece d-bodede i Baach abtract pace. By two differet proof Theorem 3 below how that i a Baach pace S a arbitrary eqece i coverget to a elemet of S if (a by the firt ieqality i ( the give eqece i boded ad moreover (b by the ecod ieqality i ( a mch more deep bodede coditio i met amely that the give eqece by ay bod D of the give eqece i ch that eqece ( i.e. (5 be boded (ad vice-vera. I particlar Theorem 3 deal with erie defied i the ormed or Baach abtract pace ( S that i ( h= whoe eqece of partial m i (2 ( h ( h x x h =. ( h x = =. h By it firt proof Theorem 3 how that if ( i atified by eqece (2 the erie ( atifie the Cachy eceary ad fficiet coditio for erie mmability that i (3 + h= + x ( h ( ( < ε > η ε ε 0 = 2 5

6 Lic Paper. 69 dicembre 999 or which amot to the ame thig eqece (2 atifie the Cachy eceary ad fficiet coditio for eqece covergece that i (4 + < ε > η( ε ε ( 0 = 0 for ome fctio η(ε e.g. [Die pp ] [Tré p. 52] [o pp ] [Ba pp. 9 53]. A a by-prodct Theorem 3 will alo prove that ( implie the eceary coditio for erie mmability i.e. i particlar that (5 x ( D x 2D / = 3 ( = cot. > 0 D. By it ecod proof which i baed o the fficiecy part of Lemma 2A give i the Appedix Theorem 3 how that the Cachy coditio (4 hold for ay d-boded eqece. Th if ch a eqece i the partial m eqece ( the the Cachy coditio (3 for erie obvioly follow. The ecod proof will be give i the Appedix at the ed of the paper. We ow tate Theorem 3 ad give it firt proof. THEOREM 3. (Seqece/erie covergece/mmability by d-bodede i ormed or Baach abtract pace. Coider the ormed (Baach abtract pace ( S. The (i a erie defied i ( S i mmable (to a elemet of S if the eqece of it partial m i d-boded (ii a eqece defied i ( S i a Cachy eqece (i coverget to a elemet of S if it i d-boded. Firt proof of Theorem 3. (i: (i.a we prove that if ( S i a ormed pace ad eqece (2 i ch that ( hold the the Cachy coditio (3 for erie mmability hold a well; ideed by ( ad Theorem (7 hold; th by (7 the firt ieqality i ( a property of the orm ad idetitie (A of Lemma A i the Appedix we get (5 that i ( x x + ( / 2D/ = 3 D ( = cot. > 0 frther by (5 ad the triaglar property of the orm we have (6 + h= + x D ; ( h + ( h + x 2D / ( h < 2D / = h= + that a give (3 with h= + (7 ( 2 ( 0 ideed by iverio of (7 we obtai η ε = D ε ε = ; 6

7 Roberto D'Agiò O boded eqece ad erie covergece i Baach abtract pace 2D η (8 = ε η = η( ε ε ( = from which we have 2D 0 (9 < ε > η = η( ε ε ( 0 = 0 th by (8-(9 we have that (6 a give h + x < 2D / < 2D / η = ε > η ε ε h= + ( ( ( 0 = which i (3 with (7. (i.b if ( S i a Baach pace by (i.a ad completee the aert i proved. (ii: (ii.a we prove that if ( S i a ormed pace the for ay eqece ch that ( hold the Cachy coditio (4 for eqece covergece hold a well; ow ay give eqece ; i a liear pace S ca be repreeted a the eqece of partial m of ome ad iqe erie; ideed give a eqece (20 ( ( = x hece by (20 the give eqece (2 x = we ca et = 3 ; ca be repreeted i the partial m form ( h = x = = ; h th if the eqece i ch that ( hold the by (2 ad propoitio (i.a above it atifie (3 or which amot to the ame thig it atifie (4; (ii.b if ( S i a Baach pace by (ii.a ad completee the aert i proved Notice that i the light of Lemma ad Theorem Theorem 3 above (ay it verio (A ha two frther eqivalet ad meaigfl verio (B-(C. Ideed o the oe had (B Theorem 3 prove that if a (poibly diverget eqece i boded the ch a eqece i CH- coverget (i.e. it coverge i the ee of Ceàro-Hölder amely the eqece (3 of the CH- mea of coverge i the ordiary ee. O the other had (C Theorem 3 prove that ay give eqece coverge i the ordiary ee if it derlyig eqece i boded i the ordiary ee. Th by a approach actally deiged for diverget eqece (or erie i.e. the Ceàro-Hölder method Theorem 3 cceed i provig a fficiet coditio for the ordiary covergece of eqece (or mmability of erie throgh the ordiary bodede of the derlyig eqece. Moreover a ca be eaily ee verio (B agree with a claical relt of the covergece ad mmability theory ee 7

8 Lic Paper. 69 dicembre 999 e.g. [o pp. 73-4] [Zyg p. 74] [Toe] [Meg pp ] [Woj p. 57]. However a frther iteretig featre of the covergece ad mmability relt give by Theorem 3 i that Cachy coditio Baach completee ad the Ceàro-Hölder mea (3 excepted it firt proof above a well a it ecod proof i the Appedix do ot ivolve ay deep relt or cotrct of the exitig covergece ad mmability theory. The ext relt prove that at variace with ordiary bodede d-bodede i more retrictive tha covergece itelf. I fact a proved by Theorem 4 below a coverget eqece i d-boded if ad oly if a particlar coditio i met. THEOREM 4. Coider the ormed (Baach abtract pace ( S. The a Cachy eqece i d-boded if ad oly if d η + (22 ε( η = = ( d = cot. > 0 where ε(η i the ivere fctio of η(ε i the Cachy coditio (4. Proof. Sfficiecy. For ay Cachy eqece (23 d η + d + (4 hold ad by (2 it give + < = ( ( ε (24 η( ε = ( ε a ε 0 d ; η ε ε 0 = th (23-(24 give ieqalitie (5A (ee Lemma 2A i the Appedix with d=2d; the by the eceity part of Lemma 2A (25 d / i d-boded that i ( d / 2 = ( d / 2 = cot. > 0 + Neceity. Coverely if a Cachy eqece. i ch that (25 hold the by the ecod proof of Theorem 3(ii i the Appedix we get (5A which i (4 with (2 or (2A- (3A ad d=2d The ecod proof of Theorem ow follow a a coeqece of Theorem 3 ad the oly if part of Theorem 4. Notice that the firt proof of Theorem 3 doe ot mae e of Theorem. A for the proof of the oly if part of Theorem 4 it deped o the ecod proof of Theorem 3 which (a how i the Appedix deped o the fficiecy part of Lemma 2A whoe proof doe ot mae e of Theorem either. For the reader coveiece we repeat below mtati mtadi the tatemet of Theorem. 8

9 Roberto D'Agiò O boded eqece ad erie covergece i Baach abtract pace THEOREM. Coider the abtract ormed pace ( S. The for ay eqece = (i d-bodede implie bodede (ii bodede doe ot imply d-bodede. Secod proof of Theorem. (i: if a eqece a Cachy eqece ad therefore i boded. (ii: coider a eqece i d-boded the by Theorem 3(ii i ch that (a it i a Cachy eqece ad (b (22 doe ot hold; the by (a the eqece boded ad by (b ad the oly if part of Theorem 4 it i ot d-boded i 4. Appedix. Lemma A prove idetitie (A-(2A. Idetitie (A are referred to i the firt proof of Theorem 3. Idetitie (2A are referred to i the proof of Lemma 2A whoe fficiecy part i referred to i the ecod proof of Theorem 3 at the ed of thi Appedix. LEMMA A. (Idetitie for eqece/erie i liear abtract pace. Coider erie ( i the liear abtract pace S ad the eqece of it partial m (2. The (A where (2A ( ( x = ( / = 3 x = i the derlyig eqece of eqece (2; frther for ay eqece + = + Proof of (A. Obvioly by ( we have (3A ( = 0 ( ( = + ( ( x = x = = 3. Frther by Lemma (3 ad (6 hold for ay eqece defied i S the they hold for eqece (2 a well. Th by (3 the firt eqality i (6 ad (3A we have (4A ( ( x = x ( ν ( ν / / = = ν= ν= where the firt idetity agree with (A ad the ecod idetity i (4A give that i x ( / = 3 x ( ν = ν= ( ( ν = / ν= ( / / =

10 Lic Paper. 69 dicembre 999 which by (3 i the ecod idetity i (A. Proof of (2A. By (3 we have that i or + + ( ν ( ( ν = / + ν= ν= / + + / + + ( ν + ( ν ( ( ν = ν= ν= + ν= ( / + ( = + + ( ν = / ( ν / which by (3 ad (5 give (2A ν= + ν= Lemma 2A prove ieqalitie (5A which characterie d-boded eqece. Frther the fficiecy part of Lemma 2A provide the bai for the ecod proof of Theorem 3 at the ed of thi Appedix. LEMMA 2A. (Ieqalitie that characterie d-boded eqece i ormed abtract pace. Coider the ormed abtract pace ( S ad let eqece (5A =. If ad oly if eqece + 2D + be the derlyig eqece of a atifie ( the = = 0. Proof. Sfficiecy. By ( ad a well-ow property of the orm we obtai (6A ( ( ( ( + 2D = = 0 ; + + th by (2A i orm ad (6A ieqalitie (5A hold. Neceity. Coverely if (5A hold the by (2A i orm we get (6A; frther by ettig = i (6A ad a well-ow property of the orm we have ( (7A ( ( ( ( + + 2D = ; ow we prove that if (5A ad (7A hold ad ( doe t the a cotradictio arie; ideed if ( doe ot hold the we ca et (8A D ( D+ δ = δ ( 0 + for ay fixed poitive δ; th by the ecod formla i (8A ad ettig = i (8 of Theorem we get (9A ( ( ( D + δ 2 + D = δ 0 ; + 0

11 0 Roberto D'Agiò O boded eqece ad erie covergece i Baach abtract pace the by (9A ad the firt formla i (8A we have max that i ( = max ( 2 + D ( D+ δ ( (0A max ( 0 D + 0 ( = max ( ( 2 + D δ = ( 2 D + + (A > ( δ / δ ( 0 D ; th by (0A-(A ad (7A we have the cotradictio ( 2 + 2D > ( δd / δ ( 0 D By it ecod proof which i baed o the fficiecy part of Lemma 2A Theorem 3 how that the Cachy coditio (4 hold for ay d-boded eqece (Theorem 3(ii; th if ch a eqece i the partial m eqece ( the the Cachy coditio (3 for erie obvioly follow (Theorem 3(i. For the reader coveiece we repeat below the tatemet of Theorem 3. THEOREM 3. (Seqece/erie covergece/mmability by d-bodede i ormed or Baach abtract pace. Coider the ormed (Baach abtract pace ( S. The (i a erie defied i ( S i mmable (to a elemet of S if the eqece of it partial m i d-boded (ii a eqece defied i ( S i a Cachy eqece (i coverget to a elemet of S if it i d-boded. Secod proof of Theorem 3. (ii: (ii.a we prove that if ( S i a ormed pace the for ay eqece ch that ( hold the Cachy coditio (4 for eqece covergece hold a well; ow by ( ad the fficiecy part of Lemma 2A (5A hold; th (5A a give (4 with (2A η( ε = ( 2 ε ε ( 0 = 0 ideed by iverio (2A give 2D η + D ; (3A = ε η = η( ε ε ( = ad therefore alo give 0

12 Lic Paper. 69 dicembre 999 2D < ε + (4A ( ( > η = η ε th by (3A-(4A we have that (5A a give (5A 2D + 2D η + ε 0 = 0 + < = ε > η( ε = which i (4 with (2A. (ii.b if ( S i a Baach pace by (ii.a ad completee the aert i proved. (i: (i.a let ( S be a ormed pace ad the eqece (2 of partial m of a erie ch that ( hold; therefore by propoitio (ii.a the Cachy coditio (4 for eqece i atified by the eqece ; 0 ; frthermore ice i a eqece of partial m the by ( (4 implie the Cachy coditio (3 for erie; (i.b if ( S i a Baach pace by (i.a ad completee the aert i proved 2

13 Roberto D'Agiò O boded eqece ad erie covergece i Baach abtract pace Referece. [Ba] S. Baach Théorie de Opératio Liéaire Hafer New Yor 932. [Ce] E. Ceàro Sr la Mltiplicatio de Série Blleti de Sciece Mathématiqe Darbox pp [o]. opp Ifiite Seqece ad Serie Dover New Yor 956. [Meg] R. E. Meggio A Itrodctio to Baach Space Theory Spriger New Yor 998. [Str]. R. Stromberg A Itrodctio to Claical Real Aalyi Wadworth Pacific Grove 98. [Toe] O. Toepliz Über allgemeie lieare Mittelbildge Prace Mat. Fiz. 2 pp [Tré] V. Tréogie Aalye foctioelle MIR Moco 985. [Wid] D.V. Widder Advaced Calcl Dover New Yor 989. [Woj] P. Wojtazczy Baach Space for Aalyt Cambridge Uiverity Pre Cambridge 99. [Zyg] A. Zygmd Trigoometric Serie Cambridge Uiverity Pre Cambridge