EQUIVALENCE OF n-norms ON THE SPACE OF p-summable SEQUENCES

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1 J Indones Math Soc Vol xx, No xx (0xx), xx xx EQUIVALENCE OF n-norms ON THE SPACE OF -SUMMABLE SEQUENCES Anwar Mutaqin 1 and Hendra Gunawan 5 1 Deartment of Mathematics Education, Universitas Sultan Ageng Tirtayasa, Serang, Indonesia, anwarmutaqin@gmailcom Deartment of Mathematics, Institut Tenologi Bandung, Bandung, Indonesia, hgunawan@mathitbacid Abstract We study the relation between two nown n-norms on l, the sace of -summable sequences One n-norm is derived from Gähler s formula [3], while the other is due to Gunawan [6] We show in articular that the convergence in one n-norm imlies that in the other The ey is to show that the convergence in each of these n-norms is equivalent to that in the usual norm on l Key words: n-normed saces, -summable sequence saces, n-norm equivalence Abstra Dalam maalah ini dielaari aitan antara dua norm-n di l, ruang barisan summable- Norm-n ertama dieroleh dari rumus Gähler [3], sementara norm-n edua dierenalan oleh Gunawan [6] Ditunuan antara lain bahwa eonvergenan dalam norm-n yang satu mengaibatan eonvergenan dalam normn lainnya Kuncinya adalah bahwa eonvergenan dalam masing-masing norm-n tersebut setara dengan eonvergenan dalam norm biasa di l Kata unci: ruang norm-n, ruang barisan summable-, esetaraan norm-n 1 Introduction In [6], Gunawan introduced an n-norm on l (1 ), the sace of -summable sequences (of real numbers), given by the formula x 1,, x n := 1 x 11 x n1 1/ n! 1 n x 1n x nn 000 Mathematics Subect Classification: Received: dd-mm-yyyy, acceted: dd-mm-yyyy 1

2 A Mutaqin and H Gunawan for 1 <, and x 1,, x n = su 1 su su n x 11 x 1n x n1 x nn, where x i = (x i ), i = 1,, n For =, the above formula may be rewritten as 1/ x 1, x 1 x 1, x n x 1,, x n =, x n, x 1 x n, x n where x i, x denotes the usual inner roduct on l Here x 1,, x n reresents the volume of the n-dimensional aralleleied sanned by x 1,, x n in l In general, an n-norm on a real vector sace X is a maing,, : X n R which satisfies the following four conditions: (N1) x 1,, x n = 0 if and only if x 1,, x n are linearly deendent; (N) x 1,, x n is invariant under ermutation; (N3) αx 1,, x n = α x 1,, x n for α R; (N4) x 1 + x 1, x,, x n x 1, x,, x n + x 1, x,, x n The theory of n-normed saces was develoed by Gähler in 1969 and 1970 [3, 4, 5] The secial case where n = was studied earlier, also by Gähler, in 1964 [] Related wor may be found in [1] For more recent wors, see [7, 8, 10] If X is equied with a norm, then according to Gähler, one may define an n-norm on X (assuming that X is at least n-dimensional) by the formula f 1 (x 1 ) f 1 (x n ) x 1,, x n := su f i X, f i 1 i = 1,,n f n (x 1 ) f n (x n ) Here X denotes the dual of X, which consists of bounded linear functionals on X For X = l (1 < ), we now that X = l with = 1 In this case the above formula reduces to x1 z 1 x1 z n x 1,, x n := su, z i l, z i 1 i = 1,,n xn z 1 xn z n where denotes the usual norm on l and each of the sums is taen over N Thus, on l, we have two definitions of n-norms, one is due to Gunawan and the other is derived from Gähler s formula For =, one may verify that the two n-norms are identical The urose of this aer is to study the relation between the two n-norms on l for 1 < In articular, we shall show that the two n-norms are wealy equivalent, that is, the convergence in one n-norm imlies that in the other Here

3 Equivalence of n-norms on the sace of -summable sequences 3 a sequence (x(m)) in an n-normed sace (X,,, ) is said to converge to x X if x(m) x, x,, x n 0 as m, for every x,, x n X For convenience, we rove the result for n = first, and then extend it to any n Main Results Recall that Gunawan s definition of -norm on l (1 ) is given by x, y = 1 x y x y 1/ if 1 <, and { x, y = su su x y x y } Meanwhile, Gähler s definition is given by x, y = su z,w l, z, w 1 x z x w y z y w By the same tric as in [6], one may obtain x, y 1 = su z,w l, z, w 1 x y x y z w z w From the last exression, we have the following fact Fact 1 The inequality x, y 1/ x, y holds for every x, y l Proof By Hölder s inequality for = 1, we have 1 x y x y z w z w 1 x y 1 z w x y z w 1/ 1/

4 4 A Mutaqin and H Gunawan Now, observe that z w z w 1/ [ z w + z w ] z w = z w 1/ + 1/ z w 1/ But for z, w 1 we have 1 z w z w 1/ 1 (1/ ) = 1/ This roves the inequality Note that for = 1, Hölder s inequality gives 1 x x y y z z w w x, y 1 z, w But z, w z w (see [6]), and so taing the suremum over z and w 1, we get x, y 1 x, y 1 Corollary If (x(m)) converges in,, then it also converges (to the same limit) in, We shall show next that the convergence in, also imlies the convergence in, We do so by showing that: (1) the convergence in, imlies that in, and () the convergence in imlies that in, The second imlication is already roved in [6] (using the inequality x, y 1 (1/) x y ) Hence it remains only to show the first imlication Theorem 3 If (x(m)) converges in,, then it also converges (to the same limit) in Proof Let (x(m)) be a sequence in l which converges to x l in, Then, for any ɛ > 0, there exists an N N such that for m N we have 1 x (m) x x (m) x y y z z w w < ɛ for every y l and z, w l with z, w 1 [Notice here that, for each m, we have x(m) = (x (m)) l ] In articular, if we tae y := (1, 0, 0, ), z = (z )

5 Equivalence of n-norms on the sace of -summable sequences 5 with z := sgn(x(m) x) x(m) x 1 and w := (1, 0, 0, ), then we have x(m) x 1 x (m) x = x(m) x 1 [Here we are handling only the case where x(m) x 0] Next, if we tae y := (0, 1, 0, ), z = (z 1, 0, 0, ) with z 1 := sgn(x 1(m) x 1 ) x 1 (m) x 1 1 and w := (0, 1, 0, ), then we have Adding u, we get x(m) x = x 1 (m) x 1 x(m) x 1 =1 This shows that (x(m)) converges to x in < ɛ < ɛ x (m) x x(m) x 1 x(m) x 1 < ɛ Corollary 4 A sequence is convergent in, if and only if it is convergent (to the same limit) in, All these results can be extended to n-normed saces for any n As an extension of Fact 1, we have: Fact 5 The inequality x 1,, x n (n!) 1/ x 1,, x n holds for every x 1,, x n l Corollary 6 If (x(m)) converges in,,, then it converges (to the same limit) in,, Analogous to Theorem 3, we have: Theorem 7 If (x(m)) converges in,,, then it also converges (to the same limit) in Proof Let (x 1 (m)) be a sequence in l which converges to x 1 = (x 11, x 1, ) l in,, Then, for any ɛ > 0, there exists an N N such that for m N we have 1 x 11 (m) x 11 x 1n (m) x 1n z 11 z 1n n! < ɛ 1 n x n1 x nn z n1 z nn for every x,, x n l and z 1,, z n l with z 1,, z n 1 Now, tae x = z := (0,, 0, 1, 0, ) for every =,, n, where 1 is (n + 1 )-th

6 6 A Mutaqin and H Gunawan term and z 1 = (z 11, z 1, ) l we have 1=n with z 1 := sgn(x1(m) x1) x1(m) x1 1, then x 1 (m) x 1 1 x 11 (m) x 11 x 1 (m) x 1 1 Next, if we tae x = z := (0,, 0, 1, 0, ) for every =,, n, where 1 is -th term, and z 1 := (z 11, 0, 0, ) with z 11 := sgn(x 11(m) x 11 ) x 11 (m) x 11 1, then x 1(m) x 1 1 we have x 11 (m) x 11 x 1 (m) x 1 1 < ɛ Similarly, if we alter the osition of the entry 1 in x and z for =,, n, and change the nonzero entry of z 1 accordingly, then we can get and so on until Adding u, we get x 1 (m) x 1 x 1 (m) x 1 1 < ɛ x1(n 1) (m) x 1(n 1) x 1 (m) x 1 = x 1 (m) x =1 This shows that (x(m)) converges to x in < ɛ < ɛ x 11 (m) x 11 x 1 (m) x 1 1 < nɛ Corollary 8 A sequence is convergent in,, if and only if it is convergent (to the same limit) in,, Related to the above results, one may also rove that a sequence is Cauchy in,, if and only if it is Cauchy in,, [A sequence (x(m)) in an n-normed sace (X,,, ) is Cauchy if given ɛ > 0 there exists an N N such that x(l) x(m), x,, x n < ɛ whenever l, m N, for every x,, x n X] Since (l,,, ) is a Banach sace [6], we conclude, by Theorem 7, that (l,,, ) also forms an n-banach sace 3 Concluding Remars As we have mentioned earlier, the case where = is of course secial Here, the two n-norms,, and,, are identical Indeed, by using Cauchy-Schwarz inequality (see [9]), one may obtain x 1, z 1 x 1, z n x 1,, x n = su x 1,, x n x n, z 1 x n, z n z i l, z i 1 i = 1,,n

7 Equivalence of n-norms on the sace of -summable sequences 7 By taing z 1,, z n to be the orthonormalized vectors obtained from x 1,, x n through Gram-Schmidt rocess, one can show that the above uer bound is actually attained Hence we have x 1,, x n = x 1,, x n For, things are not so simle and we have difficulties in roving the strong equivalence between the two n-norms,, and,, As a matter of fact, we do not now whether the two n-norms are strongly equivalent or not The research on this roblem, however, is still ongoing Acnowledgement The research was carried out while the first author did his master thesis at Faculty of Mathematics and Natural Sciences, Institut Tenologi Bandung References [1] CR Diminnie, S Gähler, and A White, -inner roduct saces, Demonstratio Math 6 (1973), [] S Gähler, Lineare -normierte räume, Math Nachr 8 (1964), 1 43 [3] S Gähler, Untersuchungen über verallgemeinerte m-metrische Räume I, Math Nachr 40 (1969), [4] S Gähler, Untersuchungen über verallgemeinerte m-metrische Räume II, Math Nachr 40 (1969), 9 64 [5] S Gähler, Untersuchungen über verallgemeinerte m-metrische Räume III, Math Nachr 41 (1970), 3 6 [6] H Gunawan, The sace of -summable sequences and its natural n-norms, Bull Austral Math Soc 64 (001), [7] H Gunawan, On n-inner roducts, n-norms, and the Cauchy-Schwarz inequality, Sci Math Jn 55 (00), [8] H Gunawan and Mashadi, On n-normed saces, Int J Math Math Sci 7 (001), [9] H Gunawan, O Neswan and W Setya-Budhi, A formula for angles between two subsaces of inner roduct saces, Beiträge Algebra Geom 46 (005), [10] A Misia, n-inner roduct saces, Math Nachr 140 (1989),