Journal of Inequalities and Special Functions ISSN: 2217-4303, URL: http://ilirias.com/jiasf Volume 7 Issue 4(2016), Pages 218-224.
EQUIVALENCE AMONG THREE 2-NORMS ON THE SPACE OF p-SUMMABLE SEQUENCES ¨ S ¸ UKRAN KONCA1∗ , MOCHAMMAD IDRIS2
Abstract. There are two known 2-norms defined on the space of p-summable sequences of real numbers. The first 2-norm is a special case of G¨ ahler’s formula [Mathematische Nachrichten, 1964], while the second is due to Gunawan [Bulletin of the Australian Mathematical Society, 2001]. The aim of this paper is to define a new 2-norm on `p and prove the equivalence among these three 2-norms.
1. Introduction The theory of 2-normed spaces was first introduced and developed to the theory of n-normed spaces by G¨ ahler (see, [1]-[5]) in the mid 1960’s, while that of nnormed spaces was studied later by Misiak [15]. Related works may be found in, e.g., [6]-[14], [16]-[17]. We shall study the space `pP , 1 ≤ p < ∞, containing all sequences of real p numbers x = (xj ) for which j |xj | < ∞, and usual norm defined on it is P 1 kxkp := ( j |xj |p ) p . Throughout this note, we assume that p lies in the interval 1 ≤ p < ∞ unless otherwise stated. Let n be a nonnegative integer and X be a real vector space of dimension d ≥ n (d may be infinite). A real-valued function k., ..., .k on X n satisfying the following four properties is called an n-norm on X and the pair (X, k., ..., .k) is called an n-normed space (1) (2) (3) (4)
kx1 , ..., xn k = 0 if and only if x1 , ..., xn are linearly dependent; kx1 , ..., xn k is invariant under permutation; kαx1 , ..., xn k = |α| kx1 , ..., xn k for α ∈ R; kx1 + x01 , x2 , ..., xn k ≤ kx1 , x2 , ..., xn k + kx01 , x2 , ..., xn k.
2000 Mathematics Subject Classification. 46B05, 46B20, 46A45, 46A99, 46B99. Key words and phrases. 2-normed spaces; the space of p-summable sequences; completeness; norm equivalence. c
2016 Universiteti i Prishtin¨ es, Prishtin¨ e, Kosov¨ e. Submitted November 8, 2016. Published December 4, 2016. 218
EQUIVALENCE AMONG THREE 2-NORMS ON `p
219
If X is a normed space, then, according to G¨ahler, the following formula defines an n-norm on X: f1 (x1 ) · · · f1 (xn ) G .. .. .. (1.1) kx1 , ..., xn k := sup . . . . 0 fi ∈X , kfi k≤1 fn (x1 ) . . . fn (xn ) i=1,...,n Here X 0 denotes the dual of X, which consists of bounded linear functionals on X. For X = `p , the space of p-summable sequences of real numbers, the above formula reduces to P P ··· j x1j z1j j x1j znj G . . . .. .. .. (1.2) kx1 , ..., xn kp := sup , 0 p P P zi ∈` , kzi kq ≤1 ... i=1,...,n j xnj z1j j xnj znj where k.kq denotes the usual norm on X 0 = `q and each of the sums is taken over j ∈ N. Here q denotes the dual exponent of p, so that p1 + 1q = 1 (1 < p < ∞). In 2001, Gunawan [7] defined a different n-norm on `p (1 ≤ p < ∞) by 1 x1j1 · · · xnj1 p p 1 X X H .. , .. ... abs ... kx1 , ..., xn kp := . . n! j jn x1j 1 . . . x njn n
(1.3)
where xi = (xij ), i = 1, ..., n, j ∈ N. Thus, on `p , we have two definitions of nnorms, one is derived from G¨ ahler’s formula given by equation (1.2) and the other is due to Gunawan given by equation (1.3). For p = 2, one may verify that the two n-noms are identical (see [6]). The equation (1.1) reduces to the equation (1.4) for n = 2 and x, y ∈ `p fu (x) fv (x) G (1.4) kx, yk := sup fu (y) fv (y) . fu ,fv ∈X 0 kfu k≤1, kfv k≤1
p
as
p
For X = ` and x, y ∈ ` the above formula, for convenience, can be rewritten hx, ui hx, vi G kx, ykp := sup hy, ui hy, vi q u, v ∈ ` kukq , kvkq ≤ 1
where q is the conjugate of p such that 1q = 1 − p1 (1 < p < ∞) and `q is the dual of `pP , which consists of bounded linear functionals fu , P fv on `p where fu (x) = ∞ ∞ p q hx, ui := k=1 xk uk (x ∈ ` and u ∈ ` ), fv (x) = hx, vi := k=1 xk vk (x ∈ `p and q q v ∈ ` ) and k·kq is usual norm on ` . For n = 2 and x, y ∈ `p the equation (1.3) reduces to the following equation:
H
kx, ykp
∞ ∞ x 1 XX := abs i yi 2 i=1 j=1
1 p p xj . yj
Wibawah-Kusuma and Gunawan [17] proved that Gunawan’s and G¨ahler’s nnorms on `p are strongly equivalent.
220
S. KONCA, M. IDRIS
Lemma 1.1. (Theorem 2.3, [17]) For any x1 , ..., xn ∈ `p we have 1
(n!) p
−1
H
1
G
H
kx1 , ..., xn kp ≤ kx1 , ..., xn kp ≤ (n!) p kx1 , ..., xn kp .
If we take n = 2, then the inequality given above for x, y ∈ `p reduces to 1
H
G
1
H
2 p −1 kx, ykp ≤ kx, ykp ≤ 2 p kx, ykp .
(1.5)
Lemma 1.2. (Lemma 2.1, [10]) For every x, y ∈ `p we have H
1
kx, ykp ≤ 21− p kxkp kykp . In this work, we define a new 2-norm on `p and prove the equivalence of these three 2-norms on `p . 2. Main Results Let x, y ∈ `p and `q be the dual of `p , which consists of bounded linear functionals where q is the conjugate of p such that 1q = 1− p1 (1 < p < ∞) and fu , fv on `p where P∞ P∞ fu (x) = hx, ui := k=1 xk uk (x ∈ `p and u ∈ `q ), fv (x) = hx, vi := k=1 xk vk (x ∈ `p and v ∈ `q ). Throughout the paper, by k·kp we mean the usual norm on `p . Now, we define a new formula on `p as follows: SI
kx, ykp :=
1 sup 2 q u, v ∈ ` kukq , kvkq ≤ 1
khy, ui x − hx, ui ykp + khy, vi x − hx, vi ykp
.
p The following fact tells us that the function k., .kSI p is a 2-norm on ` . p Fact 2.1. The real-valued function k., .kSI p defines a 2-norm on ` .
Proof. We need to check that k., .kSI p satisfies the four properties of a 2-norm. The ”if” part of (1), (2), (3) and (4) are obvious. To verify the ”only if” part of (1), = 0. Then from the definition of norm we have hy, ui x − assume that k., .kSI p hx, ui y = 0 and hy, vi x − hx, vi y = 0. Thus, x=
hx, ui y hy, ui
(hy, ui = 6 0)
x=
hx, vi y hy, vi
(hy, vi = 6 0) .
and
Since h., .i is bounded linear functional on `p , then
hy,ui hx,ui
and
hy,vi hx,vi
are real. For x =
hx,ui hy,ui y
(hy, ui = 6 0), if we insert the value of x in the equation hy, vi x−hx, vi y = 0, then clearly x satisfies the condition. In both cases, we conclude that x and y are linearly dependent. As a consequence of Fact 2.1 we have the following corollary. Corollary 2.2. The space `p equipped with k., .kSI p is a 2-normed space.
EQUIVALENCE AMONG THREE 2-NORMS ON `p
221
Lemma 2.3. Let 1 ≤ p < ∞. For x, y ∈ `p , we have 1
SI
H
kx, ykp ≤ 2 p kx, ykp . Proof. For x, y ∈ `p and u ∈ `q , we have the following by utilizing H¨older’s inequality ∞ P p p |hy, ui xi − hx, ui yi | khy, ui x − hx, ui ykp = i=1 p ∞ P P ∞ (xi yj − xj yi )uj = i=1 j=1 !p ∞ ∞ P P ≤ |xi yj − xj yi | |uj | i=1
j=1
∞ P ∞ P
p
|xi yj − xj yi | i=1 j=1 p p H 2 kukq kx, ykp .
≤ =
∞ P
q
pq
|uk |
k=1
From the above equation, we have SI
kx, ykp =
≤
sup u, v ∈ `q kukq , kvkq ≤ 1
1 2
1 2
sup u, v ∈ `q kukq , kvkq ≤ 1 1
khy, ui x − hx, ui ykp + khy, vi x − hx, vi ykp
1
H
1
H
2 p kukq kx, ykp + 2 p kvkq kx, ykp
H
≤ 2 p kx, ykp . Now, we need the following lemma to show the equivalence between the 2-norms H k., .kSI p and k., .kp . Lemma 2.4. For x, y ∈ `p and a, b, c, d ∈ R, we have a b kx, ykH = kax − by, −cx + dykH . abs p p c d Proof. Let x, y ∈ `p and a, b, c, d ∈ R. Then we obtain ∞ ∞ p 1 XX H p kax − by, −cx + dykp = |(axi − byi ) (−cxj + dyj ) − (axj − byj ) (−cxi + dyi )| 2 i=1 j=1 =
∞
∞
∞
∞
1 XX p |adxi yj + bcxj yi − bcxi yj − adxj yi | 2 i=1 j=1
1 XX p |ad (xi yj − xj yi ) − bc (xi yj − xj yi )| 2 i=1 j=1 p p H = |ad − bc| kx, ykp . b H H kx, ykp = kax − by, −cx + dykp . d =
a Thus, we have abs c
222
S. KONCA, M. IDRIS H Lemma 2.4 is important to obtain the equivalence between k., .kSI p and k., .kp .
Theorem 2.5. For x, y ∈ `p we have 1
H
SI
1
H
2 p −1 kx, ykp ≤ kx, ykp ≤ 2 p kx, ykp . Proof. By the equation (1.5) in Lemma 1.1, we have 2 1 H G H kx, ykp ≤ 21− p kx, ykp kx, ykp . Then 1
H
kx, ykp ≤ 2 2
−
1 2p
H
kx, ykp
12
G
kx, ykp
12 12
1
= 22
−
1 2p
hx, vi H kx, yk p hy, vi
hx, ui sup hy, ui u, v ∈ `q kukq , kvkq ≤ 1
≤2
1 2
−
1 2p
. 21
sup u, v ∈ `q kukq , kvkq ≤ 1
hy, ui abs hy, vi
hx, ui hx, vi
kx, ykH p
By Lemma 2.4, we have H
1
1
kx, ykp ≤ 2 2 − 2p
H
1
sup khy, ui x − hx, ui y, − hy, vi x + (hx, vi ykp ) 2 . q u, v ∈ ` kukq , kvkq ≤ 1
Meanwhile, Lemma 1.2 helps us to obtain the below inequality. 21 1 H kx, ykp ≤ 21− p sup khy, ui x − hx, ui ykp k− hy, vi x + hx, vi ykp u, v ∈ `q kukq , kvkq ≤ 1 1 1− p 1 1 ≤2 sup khy, ui x − hx, ui yk + khy, vi x − hx, vi yk p p 2 2 u, v ∈ `q kukq , kvkq ≤ 1 1
SI
= 21− p kx, ykp . (2.1) Hence, by Lemma 2.3 and equation (2.1) we obtain the following equivalence 1
H
SI
1
H
2 p −1 kx, ykp ≤ kx, ykp ≤ 2 p kx, ykp . G
SI
The equivalence relation between the 2-norms kx, ykp and kx, ykp can be obtained with a simple process. Theorem 2.6. For x, y ∈ `p we have 1 G SI G kx, ykp ≤ kx, ykp ≤ 2 kx, ykp 2
EQUIVALENCE AMONG THREE 2-NORMS ON `p
223
Proof. From the equation (1.5) and Theorem 2.5, we obtain 1
H
G
1
H
1
SI
H
G
2 p −1 kx, ykp ≤ kx, ykp ≤ 2 p kx, ykp ≤ 2 kx, ykp ≤ 2 p +1 kx, ykp ≤ 4 kx, ykp . Hence, we have the result.
p
Corollary 2.7. For x, y ∈ ` the three 2-norms are equivalent.
H k., .kp ,
G k., .kp
and
SI k., .kp
Proof. It is easy to see from Theorem 2.5 and Theorem 2.6.
on `p
H We know that the space `p , k., .kp is complete (see, Theorem 2.6 in [7]). Since H SI `p , k., .kp and `p , k., .kp are equivalent two 2-norms, then we come to the main result which is given by the following corollary. SI Corollary 2.8. The space `p , k., .kp is complete. In other words, it is a Banach space. SI
Proof. Let x(m) be a Cauchy sequence in `p with respect to k., .kp . Then, by H
Theorem 2.5 x(m) is Cauchy with respect to k., .kp . But we know that `p is H
complete with respect to k., .kp , and so x(m) must converge to some x ∈ `p in H
SI
k., .kp . By another application of Theorem 2.5, x(m) also converges to x in k., .kp . This shows that `p is complete with respect to the 2-norm
SI k., .kp .
3. CONCLUDING REMARKS SI
In this work, a new 2-norm k., .kp is defined on `p . We have just seen that G¨ ahler’s and Gunawan’s 2-norms on `p are (strongly) equivalent to this new 2norm on `p . If one insists to involve the functionals on `p , one may use the fact that the dual space `q is also a 2-normed space, as well as `p (1 ≤ p < ∞). Thus one may generalize this definition of 2-norm to n-norm on `p . The results obtained in this work can be extended to n-normed space. Acknowledgments. The authors would like to thank the anonymous referees for theirs comments that helped us to improve this article. References [1] S. G¨ ahler, 2-metrische r¨ aume und ihre topologische struktur, Math. Nachr. 26 (1963) 115– 148. [2] S. G¨ ahler, Lineare 2-normierte r¨ aume, Math. Nachr. 28 (1964) 1–43. [3] S. G¨ ahler, Untersuchungen ber verallgemeinerte m-metrische r¨ aume I, Math. Nachr. 40 (1969) 165–189. [4] S. G¨ ahler, Untersuchungen ber verallgemeinerte m-metrische r¨ aume II, Math. Nachr. 40 (1969) 229–264. [5] S. G¨ ahler, Untersuchungen ber verallgemeinerte m-metrische r¨ aume III, Math. Nachr. 41 (1970) 23–36. [6] S. M. Gozali, H. Gunawan, O. Neswan, On n-norms and bounded n-linear functionals in a Hilbert space, Ann. Funct. Anal. 1 (2010) 72–79. [7] H. Gunawan, The space of p-summable sequences and its natural n-norms, Bull. Austral. Math. Soc. 64 (2001) 137–147. [8] H. Gunawan, W. Setya-Budhi, Mashadi, S. Gemawati, On volume of n-dimensional parallelepipeds in `p spaces, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 16 (2005) 48–54.
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[9] H. Gunawan, Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 10 (2001) 631–639. [10] M. Idris, S. Ekariani, H. Gunawan, On the space of p-summable sequences, Mat. Vesnik 65 1 (2013) 58–63. [11] S ¸ . Konca, M. Idris, H. Gunawan, M. Ba¸sarır, p-Summable sequence spaces with 2-inner products, Contemporary Anal. Appl. Math. 3 2 (2015) 213–226. [12] S ¸ . Konca, M. Idris, H. Gunawan, A new 2-inner product on the space of p-summable sequences, J. Egyptian Math. Soc. 24 2 (2016) 244–249. [13] S ¸ . Konca, Some notes on the space of p-summable sequences, Kuwait J. Sci. (2016) accepted. [14] S ¸ . Konca, H. Gunawan, M. Ba¸sarır, Some remarks on `p as an n-normed space, Mathematical Sci. Appl. E-Notes 2 2 (2014) 45–50. [15] A. Misiak, n-inner product spaces, Math. Nachr. 140 (1989) 299–319. [16] A. Mutaqin, H. Gunawan, Equivalence of n-norms on the space of p-summable sequences, J. Indones. Math. Soc. 16 (2010). [17] R. A. Wibawa-Kusumah, H. Gunawan, Two equivalent n-norms on the space of p-summable sequences, Periodica Math. Hungar. 67 1 (2013) 63–69. Department of Mathematics, Bitlis Eren University, 13000, Bitlis, Turkey E-mail address:
[email protected] Department of Mathematics, Institute of Technology Bandung, Bandung 40132, Indonesia, (2 Permanent Address: Department of Mathematics, Lambung Mangkurat University, 70711, Banjarbaru, Indonesia). E-mail address:
[email protected],
[email protected] 