ALGEBRABILITY OF CONDITIONALLY CONVERGENT SERIES WITH CAUCHY PRODUCT

Transcription

1 ALGEBRABILITY OF CONDITIONALLY CONVERGENT SERIES WITH CAUCHY PRODUCT ARTUR BARTOSZEWICZ AND SZYMON G LA B Abstract. We show that the set of coditioally coverget real series cosidered with Cauchy product is ω, -algebrable. By FS we deote the liear space of all formal series over R. We ca cosider FS as a liear algebra with two differet products, amely for =0 x ad =0 y let x y = x y =0 poit-wise product ad x =0 =0 y = =0 =0 =0 x k y k Cauchy product. By CCS we deote the set of all coditioally coverget series. I [APS] Aizpuru et al. proved c lieability of CCS ad they cosider the algebras i FS, cosistig of elemets from CCS ad c 00. We say that subset E of some liear algebra is α, β algebrable if there is a β geerated algebra A such that A E \ {0} such that A is ot τ geerated for ay τ < β ad liear dimesio of A is equal to α. The otio of algebrability was cosidered by may authors [ACPS], [APS], [AS], [GPS], [GS], [BG]. It is easy to see that CCS is ot algebrable i FS,. However if we cosider the series of complex umbers, it appears that the set of all coditioally coverget series with poit-wise product is c, c-algebrable [BGP]. This ote is devoted to show that CCS is ω, algebrable i FS,. 99 Mathematics Subject Classificatio. Primary: 40A05; Secodary: 5A03. Key words ad phrases. algebrability, Cauchy product of series.

2 2 ARTUR BARTOSZEWICZ AND SZYMON G LA B Our mai tool will be the followig classical result by Prigsheim. family { =0 xs : s S} of series is absolutely equi-coverget if for ay ε > 0 there is N such that k= xs < ε for ay N ad s S. Theorem. [P] Let =0 a ad =0 b be coverget series. Assume that the series a 0 + a + a 2 + a 3 + a 4 + a is absolutely coverget. Suppose moreover that the family of series { } F = a ϕ b ψ : ϕ, ψ : N N with ϕ, ψ =0 is absolutely equi-coverget. The =0 c = =0 a =0 b is coverget. A. coditioally coverget series We say that series =0 a is alteratig if a 2 0 ad a 2+ 0 for ay = 0,, 2,... It is a easy observatio that the Cauchy product of two alteratig series is alteratig. Theorem 2. CCS is ω, algebrable i FS,. Proof. Put a = + for ay N. Note that the series =0 a is alteratig. Defie umbers a k iductively: a = a for ay N ad a k+ = a k m a m for ay k. We will use the well-kow fact that lim l + = γ Euler Mascheroi costat. m + The there are 0 < C < C 2 < such that C l + m + C 2 l +

3 CONDITIONALLY CONVERGENT SERIES 3 for ay. Havig this we will show iductively that C k l k + + a k Ck 2 l k + + for ay, k ad certai positive costats C k ad C k 2. This is obvious for k =. Assume that this is true for some k. The a k+ = a k m a m = a k m a m C k 2 l k m + m + + m C k 2 l k + 2Ck 2 l k m + + m = Ck 2 l k m + 2Ck 2 C 2 l k + 2Ck 2 C 2 l k Put C k+ 2 = 2C k 2 C 2. We also have a k+ = a k m a m = a k m a m m=/2 C k C k l k m + m + + m Ck l k 2 Ck l k + 2 Ck l k 2 k + 2 m /2 m m l k m + m + + m m /2 m m m + Ck C l k 2 k = + 2 = Ck C l k + 2 k + lk l k + C k C C l k + 2 k, + where 0 < C + lk + 2 l k + for ay N. Put C k+ = C k C /2 k C. m + + m Let N be such that the map lk + + is decreasig o {N, N +, N + 2,...} Note that for m N =m a ϕ ak ψ =m ϕ + Ck 2 l k ψ + ψ +

4 4 ARTUR BARTOSZEWICZ AND SZYMON G LA B =m + Ck 2 l k + = + =m C k 2 l k < for ay ϕ, ψ : N N with ϕ, ψ. See that a + a Hece usig Theorem we obtai that + 2 =0 ak+ is coverget. Let A be a sub-algebra of FS, geerated by =0 a. To ed the proof it is eough to show that the series =0 c a + c 2 a c k a k is coditioally coverget for ay atural umber k ad ay reals c,..., c k with c k 0. This follows from the fact that =0 a k = =0 ak. This series is clearly coverget as a liear combiatio of coverget series. We will show that it is ot absolutely coverget. We may assume that c k = ad k 2. Let M = max i=,2,...,k c i c k, M 2 = max i=,2,...,k C i 2. Let m 0 N be such that for ay m 0. The l + > 2k M M 2 C k a k Ck l k + = l+ Ck l k l k 2 + 2M M 2 + lk + l Therefore 2 c k 2M 2 c k c k a k c a a k c k a k + a k a 2 + a + c k 2 a k c 2 a 2 + c a c k a k + c k 2 a k c 2 a 2 + c 2 a c k a k c k a k > 2k M M 2 Ck l k 2 + C k + + c a. 2 c ka k = 2 c ka k.

5 Hece c a + c 2 a 2 =0 =m 0 CONDITIONALLY CONVERGENT SERIES c k a k =m 0 c a + c 2 a 2 c k a k c a + c 2 a c k a k c k a k c k a 2 =m 0 k =. Note that i particular we have proved that the set { =0 a k : k } is liearly idepedet. 2. Appedix Sice Prisheim s paper [P] is ot readily accessible, we reproduce here the proof of Theorem for the sake of completeess of this ote. Proof. First we will show that c 0. We have m m c 2m = a k b 2m k + a 2m k b k a m b m, Hece c 2m+ = c m a k b 2m+ k + m a k b k + m a 2m+ k b k. m a k b k + a m b m where m = max{k Z : k /2}. Sice F is absolutely equi-coverget, we fid N N with m k=n Let be such that ad a m b m < ε/5. The a k b k < ε m 5 ad a k b k < ε 5. b, b,..., b N+ < a, a,..., a N+ < N k=n a k b k < ε N 5 ad ε max i N a i 4N, ε max i N b i 4N. a k b k < ε 5

6 6 ARTUR BARTOSZEWICZ AND SZYMON G LA B Hece N a k b k + Therefore c 0. c m k=n m a k b k + m a k b k + a m b m = N a k b k + a k b k + m a k b k + a m b m < ε. k=n Recall that if the series =0 c is coverget to some C, the C = AB, where A = =0 a ad B = =0 b. Sice c 0, it is eough to show that teds to zero, if m. We have a 0 4m a 0 4m l=2m+ 4m b l +a a 0 + a b l +a D 4m = C 4m A 2m B 2m = 4m l=2m+ D 4m = 4m b l +...+a 4m 4m l=2m+ 4m k a l b k l 2m c k 2m a k 2m 2m a k b l = b l +a 4m b 0 a 0 +a +...+a 2m 2m b l +...+a 2m b 2m+ +a 2m+ b l + a 2 + a 3 4m 3 l=2m+ b l 2m 2m 2 b l +a 2m+2 b l a 2m 2 + a 2m b 2m+ + b l = b l +...+a 4m b 0 = 2m 2m 3 +a 2m +a 2m+ b l +a 2m+2 +a 2m+3 b l +...+a 4m 2 +a 4m b 0 +b + a 0 b 4m + a 2 b 4m a 2m 2 b 2m+2 + a 2m b 2m + a 2m+2 b 2m a 4m b 0 Hece D 4m m a 2k +a 2k+ + 2m a 2m b b 2m. 4m 2k+ b l l=2m+ + m a 2k b 4m 2k + a m a 2m+2k +a 2m+2k+ 2m b k. 2m 2k+ b l +

7 CONDITIONALLY CONVERGENT SERIES 7 Let G = max{ a 2k + a 2k+, b k } <. Let ε > 0 Let M N be such that the followig iequalities hold for ay m M 4m 2k+ b l < ε 4G, l=2m+ m a 2m+2k + a 2m+2k+ < ε 4G, 2m a 2m < ε 4G, a 2k b 4m 2k < ε 4. To fid such m i the last iequality oe should repeat the same reasoig as i the first part of the proof where it has bee show that c 0. Now, if m M, the D 4m < ε ad the result follows. Ackowledgemet. The authors would like to thak Jua Seoae- Sepúlveda for the careful examiatio of the paper ad for valuable suggestios. Refereces [APS] Aizpuru, A.; Pérez-Eslava, C.; Seoae-Sepúlveda, J. B. Liear structure of sets of diverget sequeces ad series. Liear Algebra Appl , o. 2-3, [APS] Aro, R. M.; Pérez-García, D.; Seoae-Sepúlveda, J. B. Algebrability of the set of o-coverget Fourier series. Studia Math , o., [AS] Aro, R. M.; Seoae-Sepúlveda, J. B. Algebrability of the set of everywhere surjective fuctios o C. Bull. Belg. Math. Soc. Simo Stevi , o., [ACPS] Aro, R. M. ; Coejero, J. A.; Peris, A.; Seoae-Sepúlveda, J. B. Ucoutably Geerated Algebras of Everywhere Surjective Fuctios. Bull. Belg. Math. Soc. Simo Stevi 7 200, [BG] Bartoszewicz, A.; G l ab, S. Strog algebrability of sets of sequeces ad fuctios, submitted. [BGP] Bartoszewicz, A.; G l ab, S.; Poreda, T. O algebrability of oabsolutely coverget series, to appear i Liear Algebra Appl. doi:0.06/j.laa

8 8 ARTUR BARTOSZEWICZ AND SZYMON G LA B [GS] García-Pacheco, F. J.; Martí, M.; Seoae-Sepúlveda, J. B. Lieability, spaceability, ad algebrability of certai subsets of fuctio spaces. Taiwaese J. Math , o. 4, [GPS] García-Pacheco, F. J.; Palmberg, N.; Seoae-Sepúlveda, J. B. Lieability ad algebrability of pathological pheomea i aalysis. J. Math. Aal. Appl , o. 2, [P] Prigsheim, A.; Ueber die Multiplicatio bedigt covergeter Reihe. Math. A , o. 3, Istitute of Mathematics, Techical Uiversity of Lódź, Wólczańska 25, Lódź, Polad address: arturbar@p.lodz.pl Istitute of Mathematics, Techical Uiversity of Lódź, Wólczańska 25, Lódź, Polad address: szymo glab@yahoo.com