Kagweo-Kyugki Math. Jour. 6 (1998), No. 2, pp. 331 339 SUBSERIES CONVERGENCE AND SEQUENCE-EVALUATION CONVERGENCE Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo Abstract. We show a series of improved subseries covergece results, e.g., i a sequetially complete locally covex space X every weakly c 0 -Cauchy series o X must be c 0 -coverget. Thus, if X cotais o copy of c 0, the every weakly c 0 -Cauchy series o X must be subseries coverget. Let X be a locally covex space. A series x j o X is said to be weakly c-coverget if for every {t j } c the series t jx j coverges i (X, weak), i.e., for every {t j } c there is a x 0 X such that t j f(x j ) = lim f( t j x j ) = f(x 0 ) for each f X, the dual of X(= the family of cotiuous liear fuctioals o X). I this case, x 0 is the weak sum of the series t j x j ad we write x 0 = w t jx j. Similarly a series x j o X is said to be c-coverget if for every {t j } c the series t jx j coverges i X. Sice c 0 c, if x j is weakly c-coverget the x j is weakly c 0 - coverget ad, by the Orlicz-Pettis theorem, x j is c 0 -coverget. Therefore we have ( ) Propositio 1. If x j is weakly c-coverget, the for all f X f(x j ) < +. Received Jue 29, 1998. 1991 Mathematics Subject Classificatio: 46A45. Key words ad phrases: subseries covergece. This research was supported by Dog-Il Scholarship ad Cultural Foudatio, 1997
332 Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo Proof. See [1], Theorem 2. Of course, if x j is (weakly) c 0 -coverget, the ( ) holds ad the coverse is true if X is sequetially complete. Note that with the orm {t j } = sup j t j, c 0, c ad l are Baach spaces. For a locally covex space X, let σ(x, X ), τ(x, X ) ad β(x, X ) deote the weak topology, the Mackey topology ad the strog topology, respectively. τ(x, X ) is just the topology of uiform covergece o weak* (σ(x, X)) compact balaced covex sets i X ad β(x, X ) is just the topology of uiform covergece o weak* bouded sets i X. If (X, ) is a Baach space, the τ(x, X ) = β(x, X ) = by the Baach-Alaoglu theorem (see [2]). For a locally covex space X (with the locally covex topology µ) ad a operator T : c X we say that T is cotiuous meas T is µ cotiuous. But µ τ(x, X ) β(x, X ) so β(x, X ) cotiuity is stroger tha cotiuity (= µ cotiuity). However, by the Helliger-Toeplitz theorem, if (Y, ) is a Baach space ad T : Y X is cotiuous, i.e., µ cotiuous, the T is β(x, X ) cotiuous because β(y, Y ) =. Thus, for T : c X, the cotiuity of T is equivalet to the β(x, X ) cotiuity. It is well kow that if x j is a (weakly) c 0 -coverget series o a locally covex space X, the lettig T {t j } = t jx j for each {t j } c 0, T is β(x, X ) cotiuous liear operator ad, hece, T is β(x, X ) cotiuous. Note that i this case the series t jx j coverges with respect to the origial topology o X ad the more strog τ(x, X ), the Mackey topology. But i the case of c-covergece, a weakly c-coverget series eed ot be c-coverget. The followig result shows that weakly c-coverget series also gives β(x, X ) cotiuous operators. Theorem 2. Let X be a locally covex space ad x j a weakly c- coverget series o X. Defie T : c X by T {t j } = w t jx j, {t j } c. The T is a cotiuous liear operator ad, hece, T is β(x, X ) cotiuous.
Subseries covergece ad sequece-evaluatio covergece 333 Proof. If {t j } c, the t j f(x j ) = lim t j f(x j ) = lim f( t j x j ) = f(w t j x j ) for all f X. Suppose that lim α {t αj } = {t j } i (c, weak). It is well kow that f c if ad oly if there exists a γ C ad a such that {γ j } l 1 = {{δ j } : δ j < + } f{s j } = γ lim s j + γ j s j j for {s j } c. Therefore, lim[γ lim t αj ] + lim t αj γ j = γ lim t j + α j α j t j γ j for every γ C ad {γ j } c. Lettig γ = 0, we the have lim α t αjγ j = t jγ j for all {γ j } l 1. Now let f X be arbitrary. By Propositio 1, {f(x j )} l 1. Therefore, lim α f(t {t αj }) = lim α f(w t αj x j ) = lim α t αj f(x j ) = t j f(x j ) = f(w t j x j ) = f(t {t j }). This shows that T is weak-weak cotiuous. By the Helliger-Toeplitz theorem ([2], P. 169, Corollary. 6), T is β(c, c ) β(x, X ) cotiuous. But β(c, c ) = so T is β(x, X ) cotiuous.
334 Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo A series x j o a locally covex space X is said to be weakly c- Cauchy if for every {t j } c, { t jx j } =1 is a Cauchy sequece i (X, weak), i.e., for each f X, { t j f(x j )} =1 = {f( t j x j )} =1 is a Cauchy sequece i C. Clearly, x j is weakly c-cauchy if ad oly if for every {t j } c ad f X the series t jf(x j ) coverges. The followig result shows that a weakly c-cauchy series o a sequetially complete locally covex space must be c 0 -coverget. Note that Baach spaces are sequetially complete locally covex spaces. Theorem 3. Let X be a sequetially complete locally covex space. If a series x j o X is weakly c-cauchy, the x j is c 0 -coverget, i.e., for each {t j } c 0 the series t jx j coverges. Proof. Suppose f(x j) = + for some f X. There is a iteger 1 > 1 such that 1 f(x j) > 1. There is a iteger 2 > 1 such that 2 f(x j) > 1 f(x j) + 2. There is a 3 > 2 such that 3 f(x j) > 2 f(x j) +3. Cotiuig this costructio we have a iteger sequece 1 = 0 < 1 < 2 < 3 < such that k+1 j= k +1 f(x j ) > k + 1, k = 0, 1, 2, 3,. Let t 1 = 0, t j = 1 k+1 sg f(x j), k < j k+1, k = 0, 1, 2, 3,. The t j 0 so {t j } c 0 c. But N N t j f(x j ) = t j f(x j ) = j=2 k=0 k+1 j= k +1 1 k + 1 (sg f(x j))f(x j ) = N k=0 1 k + 1 k+1 j= k +1 f(x j ) > N 1 = N + 1, k=0
Subseries covergece ad sequece-evaluatio covergece 335 for all N N, i.e., t jf(x j ) diverges. This cotradicts that x j is weakly c-cauchy. So f(x j) < +, foa all f X. Let For every f X, A = α j x j : N, α 1, 1 j. f( α j x j ) = α j f(x j ) f(x j ) α j f(x j ) f(x j ) < +, for all α jx j A. This shows that A is weakly bouded ad, hece, bouded by the Mackey theorem ([2], p.114, Theorem 1). Now suppose that {t j } c 0, i.e., t j 0. Without loss of geerality, we assume that for all j 0 there exists j > j 0 such that t j 0. Let U be a eighborhood of 0 X. Lettig α k = sup j k t j, α k 0. Sice A is bouded, there is a δ > 0 such that αa U for all α δ. Sice α k 0, there is a k 0 N such that if k k 0, the α k δ. Therefore, if m > k k 0, the m t j x j = α k j=k m j=k t j α k x j = α k 0x 1 + 0x 2 + + 0x k 1 + α k A U. m t j x j α k This shows that { t jx j } =1 is Cauchy ad, hece, the series t jx j coverges because X is sequetially complete. j=k
336 Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo Theorem 4. Let X be a sequetially complete locally covex space. For a series x j o X, the followig coditios are equivalet. (1) x j is a weakly ucoditioal Cauchy series, i.e., for all f X, f(x j) < +. (2) For every {t j } l, { j t jx j : N fiite} is bouded. (3) x j is c 0 -coverget, i.e., for every {t j } c 0, the series t jx j coverges. (4) x j is weakly c 0 -Cauchy, i.e., the series t jf(x j ) coverges for every {t j } c 0 ad f X. (5) x j is weakly c-cauchy, i.e., the series t jf(x j ) coverges for every {t j } c ad f X. (6) { t jx j : N, t j 1, 1 j } is bouded. Proof. By Theorem 2 of [1], (1)=(2)=(3) sice X is sequetially complete. Sice c 0 c, (5) (4). As i the proof of Theorem 3, (4) (1) (6) (3) (4). So (1)=(2)=(3)=(4)=(6) ad (5) (4). Suppose (4) holds. The (1) holds because (1)=(4), i.e., f(x j) < +, for all f X. Sice {t j } c {t j } is bouded, t j f(x j ) = t j f(x j ) sup j t j f(x j ) < +. This shows that t jf(x j ) coverges for all {t j } c. Corollary 5. If X is a sequetially complete locally covex space, the (1)=(2)=(3)=(4)=(5)=(6)=(7)=(8)=(9)=(10). (7) t jf(x j ) < +, for all {t j } c 0, f X. (8) t jf(x j ) < +, for all {t j } c, f X. (9) t jf(x j ) < +, for all {t j } l, f X. (10) t jf(x j ) coverges for every {t j } l, ad f X. Proof. {t j } l {t j sg f(x j )} l, so (9)=(10). (1) (9) (8) (7) (4) (1). Now we give the mai result of this paper.
Subseries covergece ad sequece-evaluatio covergece 337 Theorem 6. Let X be a sequetially complete locally covex space. The followig coditios are equivalet. (a) X cotais o copy of c 0. (b) Each weakly c 0 -Cauchy series o X is c-coverget, i.e., if t jf(x j ) coverges for every {t j } c 0 ad f X, the t jx j coverges for each {t j } c. (c) Each weakly c-cauchy series o X is c-coverget, i.e., if t jf(x j ) coverges for every {t j } c ad f X, the t jx j coverges for each {t j } c. Proof. (a) (b). Suppose α jf(x j ) coverges for every {α j } c 0 ad f X. Let {t j } c. The α j t j 0 for each {α j } c 0 so α jf(t j x j ) coverges for every {α j } c 0 ad f X. By theorem 4 ((3)=(4)), α jt j x j coverges for each {α j } c 0, i.e., {t j x j } CMC(X) (see [3]). Sice X cotais o copy of c 0, by Theorem 4 of [3], t jx j coverges, i.e., (b) holds. (b) (c) : c 0 c. (c) (a). Suppose X cotais a copy of c 0. Say that c 0 X. Let e j deotes the sequece that has 1 at the j-th spot ad 0 elsewhere, i.e., e j = (0,, 0, 1, 0, 0, ). For every {t j } c ad f = {α j } l 1 = c 0, t j f(e j ) = f(t j e j ) = f( t j e j ) = f(t 1, t 2,, t, 0, 0, ) = t j α j t j α j sup j t j α j sup j t j α j < +, for all N, i.e., for every {t j } c ad f c 0, t jf(e j ) coverges. However, lettig t j = 1 for all j, {t j } = {1} c but the series e j
338 Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo diverges i c 0 : e j = (0,, 0, 1, 1,,, 1, 0, 0, ) = 1 j=m for all 1 m < < +. If lim e j = x X\c 0, the lim m, j=m e j = 0. So e j diverges i X. This cotradicts (c). Corollary 7. If a sequetially complete locally covex space X cotais o copy of c 0, the every weakly c-coverget series o X is c-coverget. By Theorem 4 of [3], we have Theorem 8. Let X be a sequetially complete locally covex space. The followigs are equivalet. (1 ) X cotais o copy of c 0. (2 ) Each weakly c 0 -Cauchy series o X is bouded multiplier coverget, i.e., if t jf(x j ) coverges for every {t j } c 0 ad f X, the t jx j coverges for each {t j } l, the family of bouded umber sequeces. Proof. (1 ) (b). So if t jf(x j ) coverges for every {t j } c 0 ad f X, the {x k } CMC(X) but (1 ) CMC(X) = BMC(X) by Theorem 4 of [3]. Refereces 1. Li Roglu ad Mi-Hyug Cho, Weakly Ucoditioal Cauchy Series o Locally Covex Spaces, Northeast Math. J., 11(2) (1995), 187-190. 2. A. Wilasky, Moder Methods i Topological Vector Spaces, McGraw-Hill, New York (1978).
Subseries covergece ad sequece-evaluatio covergece 339 3. Li ad Bu, Locally Covex Spaces Cotaiig No Copy of c 0, J. Math. Aal. Appl., 172(1) (1993), 205-211. Departmet of Applied Mathematics Kum-Oh Natioal Uiversity of Techology Kumi 730-701, Korea E-mail: migo@kut.kumoh.ac.kr hthwag@kut.kumoh.ac.kr wsyoo@kut.kumoh.ac.kr