A Study on a Subset of Absolutely. Convergent Sequence Space

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1 It J Cotemp Math ciece, Vo 4, 2009, o 24, A tudy o a ubet o Aboutey Coverget equece pace U K Mira, M Mira 2, N ubramaia 3 ad P amata 4 Departmet o Mathematic, BerhampurUiverity Berhampur , Oria, Idia umaata_mira@yahoocom 2 Pricipa, Govermet ciece Coege Maagiri, Oria, Idia 3 Departmet o Mathematic, atra Uiverity Tajore , Tamiadu, Idia 4 Departmet o Mathematic, Gopapur Coege Gopapur o ea, Oria,Idia Abtract: I thi paper, we deie the ectio equece pace which i caed the ectio equece pace o ad tudy the icuio Further AK-property, Dua pace o are tudied Mathematic ubject Caiicatio: 40A05 Keyword: chauder bai, dua pace, oid, wea ad trog covergece, eparabe Itroductio ad preimiarie Let be a BK-pace We deote a the equece coitig o a thoe equece { x ( x ):( y ) }, where y x + x2 + x3 + + x, or each ixed 23,,, For a equece ( y ), we ca cacuate the equece ( x ) by

2 50 U K Mira, M Mira, N ubramaia ad P amata For ay x y, x2 y2 x y2 y, x3 y3 x x2 y3 y ( y2 y) y3 y2 y y x x, we deie { x + x + x2 + + x + x2 + + x + x + + } For a give a equece x { x } ( ) x { x, x,, x, 0,0, } x <, we deie the th ectio a the equece 2 Let ( ) δ ( 0,0,,,-,0,0,0, ), where i i the th pace ad - i the ( + ) th pace A FK-pace X i aid to have AK-property i ( ) { δ } i a chauder bai or X The pace X i aid to have AD i Φ i dee i X We ote that AK AD by [ ] For the equece pace X, we deie X { } a a : a x i coverget, or each x X We caed X α, X, X γ a the α -dua o X, -dua o X, γ -dua o X, α γ μ μ repectivey Note that X X X I X Y the Y X, or μ α, ad γ We have the oowig ow reut Lemma : (ee Theorem 727i [3] Let X be a FK-pace Φ The γ (i) X X (ii) I X ha AK, X X γ (iii) I X ha AD, X X Lemma 2 ( Page 69, 23 i [2] ):

3 ubet o aboutey coverget equece pace 5 I a Normed pace X ha a chauder bai, the X i eparabe 2 Mai Reut: Propoitio-: e I thi ectio we tudy ome o the property o { 0,0,0,,,-,0,0, } ha chauder bai amey ( e e, e, e, ), i i the () (2) Proo We ow that {, δ, }, 2 3 th pace ad - i at the ( ) th, where + or,2, δ i a chauder bai or traormatio give i the itroductio It oow that ( e e, e,, ) Theorem- ; ha AK-property e, 2 3 i a chauder bai or Proo Let x ) The ( y ) with y x + x + + x ( x ( ) x x x2 x3 x Put (,,,,,0,0, ) The T 2 ( ) x x,0,0,, x, x, x + + x + + x y + y + y + y + y + y y y 0, a + y 0, a, becuae ( y ) Thu we have ( ) 0 x x 0, or uiciety arge Hece x x ( ) 0, a, Thereore the pace ha AK Thi compete the proo () ( 2) Coroary -: The et { δ, δ, } i a chauder bai or Proo: By p59,423 i [3] Propoitio- 2: ad the icuio i trict

4 52 U K Mira, M Mira, N ubramaia ad P amata Proo Let x The y Hece have The Hece δ x y y y + y x y + x Coequety y y < But a x y y We Next we how that he above icuio i trict For thi tae the equece () (,0,0, ) The δ () ad thu we have y, y + 0, y ,, y Now, 2 () y or a Hece ( y ) doe ot ted to zero a Hece δ Thu the icuio Thi compete the proo heorem-2 : The dua o pace i Proo: A chauder bai or i ( ) where e ( ) ha i the e th pace th ad - i the ( + ) pace ad zero eewhere Let x The there exit caar α, α 2, uch that x α e i uique Now or ay bouded iear operator o we have ( x) ( α e ) α ( e ) α γ, where the umber γ e ) are uiquey determied by Ao ( γ e ), γ e ) ice i iear ad bouded ( ( ) e ( γ But e ( ) e ( 0,0,,,,0,0, ) ( ) (um o the irt term) ad e Thu

5 ubet o aboutey coverget equece pace 53 γ e γ up γ M ( ) Hece ( ) γ Thereore (2) But by Propoitio- 2, Hece A, (22) Hece rom (2) ad (22) Thi compete the proo Theorem-3 : The -dua o i Proo: By Propoitio-2 we get Hece ( ) But Hece (23) ( ) Next, et y ( ) ad ( ) x ( ) ( 0,0,,,,0, ) x y with x Tae, { x } { 0,0,0,,,0, } ( ) x, where A thi coverge to zero, ( ) Hece ( ) ( ) But (24) y ( ) ( ) ( ) Thu { y } i a bouded equece Further,a y i arbitrary i ( ) (25) ( ) From (23) ad (43) we get ( ) Thi compete the proo Propoitio-3: i oid

6 54 U K Mira, M Mira, N ubramaia ad P amata Proo: Let x y with y ( y ) o η i oid Hece ξ ( ξ ) Thereore x ( x ) compete the proo Coroary-2: I ξ with η ( y ) Hece i oid Thi, wea covergece doe ot impy trog covergece Proo: Aume that wea covergece impie trog covergece i woud have ( ) [ ee() ] But ( ) ( ) proper ubpace o Thu ( ) trog covergece i Thi compete the proo Coroary-3: ( ) μ where μ α,, γ, But The we By Propoitio 2 i a Hece wea covergece doe ot impy Proo: ha AK property, by theorem- Hece by Theorem- 739 i [3] we get ( ) ( ) But ( ) Hece (26) ( ) AK rom [3] we get ( ) ( ) γ ice AD, (27) ( ) γ Thereore By propoitio-3, we have i oid Hece by Theorem 739 i [3], We get α γ (28) ( ) ( ) α γ From (26), (27) ad (28),we have ( ) ( ) ( ) ( ) Thi compete the proo Theorem -4: Let Y be ay FK-pace Φ The Y i ad oy i ( ) { } δ i weay bouded Proo: I order to etabih the reut it i eough to etabih the oowig reut: ( ) Y Y ice ha AD ad ( ), by uig Theorem 86 i [3] we have Y

7 ubet o aboutey coverget equece pace 55 or each Y, the topoogica dua o Y δ ( ) ( ) i bouded δ Thi compete the proo ( ) ( ) ( ) { δ } i weay bouded Theequece Theorem-5: I,weay coverget equece are orm coverget Proo: Let a { a, a + a2, a + a2 + a3, } be weay coverget ad et ( a ) a iiite matrix Let u aume that A i coercive ice ( ) A be, it i a coervative matrixo the coum exit by, Theorem 36 i [3] By uig Theorem 37 i [3] a + a + a2 + a, a + a2, a + a2 + a3, im a + a2 + a3 + A ice bouded mootoic equece coverge, Propoitio-4: Thi compete the proo i ot perect a Proo: We ow that ( ) Hece ( ) ( ) But a ( ), ( ) Hece i ot perect Thi compete the proo Propoitio-5: The pace i eparabe Proo: By Propoitio,we have ha chauder bai { e, e2,, e,} Ao i a Baach pace Hece, by the Lemma- 2, it oow that i eparabe Thi compete the proo Propoitio-6: The pace i ot eparabe Proo: By Theorem 39 i [2] Propoitio-7: The pace i ot reexive Proo By Propoitio-5, we have ice proo i eparabe But, by Propoitio 2 ( ) i ot eparabe by Propoitio-6, i ot reexivethi compete the

8 56 U K Mira, M Mira, N ubramaia ad P amata Theorem-6: The pace i a ier product pace but ot a Hibert pace Proo The proo wi be etabihed by howig that the orm atiie the aw o paraeogram Let u tae {,,0, } ad y {,,0, } x The { } { 0 } x x + x + x + x + x + x { 0 0 } imiary, Coider, imiary, Now x y y { } 0 { 0 0 } { 2 2 } x+ y x + y + ( x + y ) + ( x + y ) + 2 { x y + x y ) + ( x y ) + ( x y ) + ( x y ) + ( x y ) } ( { } Thu paraeogram aw i atiied Thereore i a ier product pace For the proo o the ecod part et u uppoe that i a Hibert pace The by [2] (Theorem 466) woud atiy reexivity coditio Thi cotradict Propoitio-7 Hece i ot a Hibert pace Thi compete the proo REFERENCE [] Brow, H I, The ummabiity ied o a perect method o ummatio Joura D Aaye Mathematique, 20 (967),

9 ubet o aboutey coverget equece pace 57 [2] Erwi, Kreyzig, Itroductory Fuctioa Aayi with Appicatio, Joh Wiey & o Ic, 978 [3] Wiay, A, ummabiity through Fuctioa Aayi, North Hoad Mathematic tudie, Vo85, North-Hoad,Amterdam, 984 Received: November, 2008

ON WEAK -STATISTICAL CONVERGENCE OF ORDER

UPB Sci Bu, Series A, Vo 8, Iss, 8 ISSN 3-77 ON WEAK -STATISTICAL CONVERGENCE OF ORDER Sia ERCAN, Yavuz ALTIN ad Çiğdem A BEKTAŞ 3 I the preset paper, we give the cocept of wea -statistica covergece of