Department of Mathematics, SASTRA University, Tanjore , India

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1 Selçuk J. Appl. ath. Vl.. N.. pp. 7-4, Selçuk Jural f Applied athematics The Duble Sequeces N. Subramaia Departmet f athematics, SASTRA Uiversity, Tajre-63 4, Idia smaths@yah.cm Received Date Octber 8, 8 Accepted Date ay 3, Abstract. Let dete the space f all duble etire sequece. Let dete the space f all duble aalytic sequeces, let dete the space f all duble sequeces. This paper is devted t a study f the geeral prperties f Key wrds Duble di erece sequece spaces; etire sequece; aalytic sequece; gai sequeces; dual. athematics Subject Classi cati 4A5; 4C5; 4D5.. Itrducti Thrughut w; ad dete the classes f all, gai ad aalytic scalar valued sigle sequeces, respectively. We write w fr the set f all cmplex sequeces (x m ); where m; N; the set f psitive itegers. The, w is a liear space uder the crdiate wise additi ad scalar multiplicati. Sme iitial wrks duble sequece spaces is fud i Brmwich[4]. Later, they were ivestigated by Hardy[8], ricz[], ricz ad Rhades[3], Basarir ad Slaka[], Tripathy[], Clak ad Turkmeglu[6], Turkmeglu[], ad may thers. Let us de e the fllwig sets f duble sequeces u (t) (x m ) w sup m;n jx m j tm < ; C p (t) (x m ) w p lim m;! jx m j tm fr sme C ; C p (t) (x m ) w p lim m;! jx m j tm ; L u (t) (x m ) w P P m jx mj tm < ; C bp (t) C p (t) T u (t) adc bp (t) C p (t) T u (t) ;

2 where t (t m ) is the sequece f strictly psitive reals t m fr all m; N ad p lim m;! detes the limit i the Prigsheim s sese. I the case t m fr all m; N; u (t) ; C p (t) ; C p (t) ; L u (t) ; C bp (t) ad C bp (t) reduce t the sets u ; C p ; C p ; L u ; C bp ad C bp ; respectively. Nw, we may summarize the kwledge give i sme dcumet related t the duble sequece spaces. Gkha ad Clak [7,8] have prved that u (t) ad C p (t) ; C bp (t) are cmplete pararmed spaces f duble sequeces ad gave the ; ; duals f the spaces u (t) ad C bp (t) Quite recetly, i her PhD thesis, Zelter [9] has essetially studied bth the thery f tplgical duble sequece spaces ad the thery f summability f duble sequeces. ursalee ad Edely [3] have recetly itrduced the statistical cvergece ad Cauchy fr duble sequeces ad give the relati betwee statistical cverget ad strgly Cesar summable duble sequeces. Nextly, ursalee [3] ad ursalee ad Edely [3] have de ed the almst strg regularity f matrices fr duble sequeces ad applied these matrices t establish a cre therem ad itrduced the cre fr duble sequeces ad determied thse fur dimesial matrices trasfrmig every buded duble sequeces x (x jk ) it e whse cre is a subset f the cre f x re recetly, Altay ad Basar [33] have de ed the spaces BS; BS (t) ; CS p ; CS bp ; CS r ad BV f duble sequeces csistig f all duble series whse sequece f partial sums are i the spaces u ; u (t) ; C p ; C bp ; C r ad L u ; respectively, ad als examied sme prperties f thse sequece spaces ad determied the duals f the spaces BS; BV; CS bp ad the (#) duals f the spaces CS bp ad CS r f duble series. Quite recetly Basar ad Sever [34] have itrduced the Baach space L q f duble sequeces crrespdig t the well-kw space `q f sigle sequeces ad examied sme prperties f the space L q Quite recetly Subramaia ad isra [35] have studied the space (p; q; u) f duble sequeces ad gave sme iclusi relatis. We eed the fllwig iequality i the sequel f the paper. Fr a; b; ad < p < ; we have () (a + b) p a p + b p The duble series P m; x m is called cverget if ad ly if the duble sequece (s m ) is cverget, where s m P m; i;j x ij(m; N) (see[]). A sequece x (x m )is said t be duble aalytic if sup m jx m j m+ < The vectr space f all duble aalytic sequeces will be deted by.a sequece x (x m ) is called duble etire sequece if jx m j m+! as m;! The duble etire sequeces will be deted by. A sequece x (x m ) is called duble gai sequece if ((m + )! jx m j) m+! as m;! The duble gai sequeces will be deted by. Let dete the set f all ite sequeces. Csider a duble sequece x (x ij ) The (m; ) th secti x [m;] f the sequece is de ed by x [m;] P m; i;j x ij ij fr all m; N ; where ij detes

3 the duble sequece whse ly zer term is a i the (i; j) th place fr each i; j N A FK-space(r a metric space)x is said t have AK prperty if ( m ) is a Schauder basis fr X. Or equivaletly x [m;]! x. A FDK-space is a duble sequece space edwed with a cmplete metrizable; lcally cvex tplgy uder which the crdiate mappigs x (x k )! (x m )(m; N) are als ctiuus. If X is a sequece space, we give the fllwig de itis (i)x is the ctiuus dual f X; (ii)x a (a m ) P m; ja m x m j < ; fr each x X ; (iii)x a (a m ) P m; a m x m is cveget; fr each x X ; (iv)x a (a m ) sup m P ;N m; a mx m < ; fr each x X ; (v)let X beaf K space ; the X f f( m ) f X ; (vi)x a (a m ) sup m ja m x m j m+ < ; X X ; X are called (rkthe T eplitz)dual f X; (r geeralized Kthe T eplitz) dual fx; dual f X; dual f X respectivelyx is de ed by Gupta ad Kampta [4]. It is clear that X X ad X X ; but X X des t hld, sice the sequece f partial sums f a duble cverget series eed t t be buded. The ti f di erece sequece spaces (fr sigle sequeces) was itrduced by Kizmaz [36] as fllws Z () fx (x k ) w (x k ) Zg fr Z c; c ad `; where x k x k x k+ fr all k N Here w; c; c ad ` dete the classes f all, cverget,ull ad buded sclar valued sigle sequeces respectively. The abve spaces are Baach spaces rmed by kxk jx j + sup k jx k j Later the ti was further ivestigated by may thers. We w itrduce the fllwig di erece duble sequece spaces de ed by Z () x (x m ) w (x m ) Z where Z ; ad x m (x m x m+ ) (x m+ x m++ ) x m x m+ x m+ + x m++ fr all m; N

4 . Lemma As i sigle sequeces (see [3, Therem 7.7]). Let X be a FK-space The (i) X X f ; (ii) If X has AK, X X f ; (iii) If X has AD, X X 3. De itis ad Prelimiaries Let w dete the set f all cmplex duble sequeces. A sequece x (x m ) is said t be duble aalytic if sup m jx m j m+ < The vectr space f all prime sese duble aalytic sequeces will be deted by A sequece x (x m ) is called prime sese duble etire sequece if jx m j m+! as m;! The duble etire sequeces will be deted by The space ad is a metric space with the metric () d(x; y) sup m jx m y m j m+ m; ; ; 3; frall x fx m g ad y fy m gi A sequece x (x m ) is called prime sese duble gai sequece if ((m + )! jx m j) m+! as m;! The duble gai sequeces will be deted by The space is a metric space with the metric (3) d(x; e y) supm ((m + )! jx m y m j) m+ m; ; ; 3; frall x fx m g ad y fy m gi De e the sets x w jxmj m+ x w sup m jxmj m+! (m;! ) fr sme > < fr sme > The space is a metric space with the metric d (x; y) if > sup m jxm y mj m+ ad the space d (x; y) if is a metric space with the metric > sup m jxm y mj m+ m; ; ; 3; 4. ai Results 4.. Prpsiti

5 Prf Let x The we have ((m + )! jx m j) m+! as m;! Here, we get jx m j m+! as m;! Thus we have x ad s 4.. Prpsiti 6 Prf Let y (y m ) be a arbitrary pit i If y is t i ; the fr each atural umber p, we ca d a idex m p p such that ymp (4) p m p+ p! > p; (p ; ; 3; ) De e x fx m g by (5) xm ( p ; m+ fr (m; ) (m p ; p ) fr smep N ; therwise The x is i ; but fr i itely m; jym x m j (6) > Csider the sequece z fz m g ; where (7) s The z is a pit f X (6), P P zmy m X m xm ; Als, P P des t cverge zm z (8) ) X X x m y m diverges x xm s with z m Hece, z is i ; but, by Thus, the sequece y wuld t be i This ctradicti prves that (9) If we w chse id; where id is the idetity ad y x ad y m x m (m > ) fr all ; the bviusly x ad y ; but () X X x m y m Hece; y m

6 Frm (9) ad (), we are grated Prpsiti The dual space f is Prf First, we bserve that ; by Prpsiti 4.. Therefre But 6 ; by Prpsiti 4.. Hece () Next we shw that Let y (y m ) Csider f (x) P m P x my m with x (x m ) x [( m m+ ) ( m+ m++ )] ; ; ; ; ; ; ; ; B ; ; (m+)! ; (m+)! ; ; ; ; ; A ; ; ; ; ; ; ; ; B ; ; (m+)! ; (m+)! ; ; ; ; ; A ((m + )! jx m j) m+ ; ; ; ; ; ; ; ; ; ; (m+)! ; (m+)! ; B; ; ; (m+)! ; C ; ; ; ; A Hece cverges t zer. Therefre [( m m+ ) ( m+ m++ )] Hece d (( m m+ ) ( m+ m++ ) ; ) But jy m j kfk d (( m m+ ) ( m+ m++ ) ; ) kfk < fr

7 each m; Thus (y m ) is a duble buded sequece ad hece a aalytic sequece. I ther wrds y But y (y m ) is arbitrary i Therefre () Frm () ad () we get 4.4. Prpsiti has AK Prf Let x (x m ) ad take the [m; ] th sectial sequece f x We have d x; x [r;s] sup m ((m + )! jx m j) m+ m r; s! as [r; s]! Therefre x [r;s]! x i as r; s! Thus has AK Prpsiti is slid Prf Let jx m j jy m j ad let y (y m ) We have ((m + )! jx m j) m+ ((m + )! jy m j) m+ But ((m + )! jy m j) m+ ; because y That is ((m + )! jy m j) m+! ) ((m + )! jx m j) m+! as m;! Therefre x (x m ) This cmpletes the prf Prpsiti dual f is Prf Let y dual f The jx m y m j m+ fr sme cstat > ad fr each x Therefre jy m j m+ fr each m; by takig ; ; ; ; ; ; ; ; x m B ; ; (m+)!; ; ; ; ; ; A This shws that y The (3)

8 O the ther had, let y Let > be give. The jy m j < m+ fr each m; ad fr sme cstat > But x Hece ((m + )! jx m j) < m+ fr each m; ad fr each > i.e jx m j < m+ (m+)! Hece ) y jx m y m j jx m j jy m j < (4) Frm (3) ad (4) we get 4.7. Prpsiti m+ (m+)! m+ ()m+ (m+)! Let dete the dual space f The we have Prf We recall that ; ; ; ; ; ; ; ; x m B ; ; (m+)!; ; ; ; ; ; A with (m+)! i the (m; ) th psiti ad zer ther wise, with x m ; ((m + )! jx m j) m+ ; ; ; ; + m+ B m+ ; ; (m+)! (m+)! ; ; m++ m+ ; ; ; ; m++ A ; ; ; ; ; ; ; ; B; ; m+ ; ; ; ; ; ; A

9 which is a duble sequece. Hece m Let us take f (x) P P m x my m with x ad f Take x (xm ) m The jy m j kfk d ( m ; ) < fr each m; Thus (y m ) is a buded sequece ad hece a duble aalytic sequece. I ther wrds y Therefre 4.8. Prpsiti Prf Step Let (x m ) ad let (y m ) The we get jy m j m+ fr sme cstat > Als Hece (x m ) ) ((m + )! jx m j) m+ ) jx m j m+ m+ (m+)! P P m jx my m j P P m jx mj jy m j < P P m < P m P m+ m+ m+ (m+)! m+ (m+)! < Therefre, we get that (x m ) ad s we have (5) Step Let (x m ) This says that (6) ) X m X jx m y m j < fr each (y m ) Assume that (x m ) ; the there exists a sequece f psitive itegers (m p + p ) strictly icreasig such that Take x mp+ p > ; (p ; ; 3; ) mp+p (m + )!

10 y mp; p mp+p (m + )! (p ; ; 3; ) ad The (y m ) But P m P jx my m j P P y m therwise p x mp p y mp p > We kw that the i ite series +++ diverges. Hece P m P jx my m j diverges. This ctradicts (6). Hece (x m ) Therefre (7) Frm (5) ad (7) we get 4.9. De iti Let p (p m ) is a duble sequece f psitive real umbers. The (p) x (x m ) ((m + )! jx m j) m+ p m! as m;! suppse that p m is a cstat fr each m; the (p) 4.. Prpsiti Let p m q m ad let qm p m be buded. The (q) (p) Prf Let (8) x (q) Therefre we have (9) ((m + )! jx m j) m+ q m! as m;! ((m + )! jx m j) m+ q m ad m p m q m Sice p m q m ; Let t m we have m Take < < m De e ( t m ; if (t m ) () u m ; if (t m < ) ; v m ( ; if (t m ) t m ; if (t m < )

11 t m u m + v m ; t m m u m m t m + v m u m t m ; v m m u m m + vm m Nw it fllws that u m Sice t m m u m m ((m + )! jx m j) m+ q m m ((m + )! jx m j) m+ q m + vm m ; the t m m ) ((m + )! jx m j) m+ q m pmq m ((m + )! jx m j) m+ q m ) ((m + )! jx m j) m+ q m pm ((m + )! jx m j) m+ q m But ((m + )! jx m j) m+ q m! as m;! (by (4)). Therefre ((m + )! jx m j) m+ p m! as m;! Hece () x (p) Frm (8) ad () we get (q) (p) 4.. Prpsiti (a) Let < ifp m p m The (p) (b) Let p m supp m < The (p) Prf (a) Let x (p) () ((m + )! jx m j) m+ p m! as m;! Sice < ifp m p m ; (3) ((m + )! jx m j) m+ ((m + )! jx m j) m+ p m Frm () ad (3) it fllws that (4) x Thus (p) This cmpletes the prf. Prf (b) Let p m fr each m ad supp m < ad let x (5) ((m + )! jx m j) m+! as m;! Sice p m supp m < ; we have (6) ((m + )! jx m j) m+ p m ((m + )! jx m j) m+

12 ((m + )! jx m j) m+ p m! as m;! (by usig (5)). Therefre x (p) 4.. Prpsiti Let < p m q m < fr each m; The x (p) x (q) Prf Let x (p) (7) ((m + )! jx m j) m+ p m! as m;! This implies that ((m + )! jx m j) m+ fr su cietly large m; We get (8) ((m + )! jx m j) m+ q m ((m + )! jx m j) m+ p m ) ((m + )! jx m j) m+ q m! as m;! (by usig (7)). Sice x (q) ; hece x (p) (q) 4.3. Prpsiti fr ; ; ; f Prf Step has AK by Prpsiti (4.4). Hece by Lemma () (ii), we get f But Hece (9) Step Sice AK implies AD, hece by Lemma (iii) we get Therefre (3) Step 3 is rmal by Prpsiti (4.5). Hece, [4, Prpsiti.7], we get (3) Frm (9),(3) ad (3), we have f 5. Ackwledgemet I wish t thak the referees fr their several remarks ad valuable suggestis that imprved the presetati f the paper ad als I wish t thak Prf. Dr. Ahmet Sia Cevik,Departmet f athematics, Faculty f Sciece, Selcuk

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