Two equivalent n -norms on the space of p -summable sequences
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TWO EQUIVALENT n-NORMS ON THE SPACE OF p-SUMMABLE SEQUENCES R. A. Wibawa-Kusumah1 and H. Gunawan2 1
Department of Mathematics, Institut Teknologi Bandung Bandung, Indonesia
2
Department of Mathematics, Institut Teknologi Bandung Bandung, Indonesia E-mail: [email protected] (Received June 29, 2011; Accepted September 11, 2012) [Communicated by D´enes Petz]
Abstract We prove the (strong) equivalence between two known n-norms on the space ℓp of p-summable sequences (of real numbers). The first n-norm is derived from G¨ ahler’s formula [3], while the second is due to Gunawan [7]. The equivalence is proved by using the properties of the volume of n-dimensional parallelepipeds in ℓp .
1. Introduction In the 1960’s, S. G¨ahler [2], [3], [4], [5] developed the theory of n-normed spaces. An n-norm on a real vector space X (of dimension at least n) is a mapping k·, . . . , ·k: X n → R which satisfies the following four conditions: (1.1) kx1 , . . . , xn k = 0 if and only if x1 , . . . , xn are linearly dependent; (1.2) kx1 , . . . , xn k is invariant under permutation; (1.3) kαx1 , . . . , xn k = |α| kx1 , . . . , xn k for α ∈ R; (1.4) kx1 + x′1 , x2 , . . . , xn k ≤ kx1 , x2 , . . . , xn k + kx′1 , x2 , . . . , xn k. The pair (X, k·, . . . , ·k) is called an n-normed space. Note that in this space, we have kx1 + y, x2 , . . . , xn k = kx1 , x2 , . . . , xn k for any y = c2 x2 + · · · + cn xn . See [1], [8], [9], [10], [13], [17] for various results on n-normed spaces. See also [14] for a related topic. If X is a normed space, then, according to G¨ ahler, the following formula defines an n-norm on X: f1 (x1 ) · · · f1 (xn ) .. .. ′ .. kx1 , . . . , xn k := sup . . . . fi ∈X ′ , kfi k≤1 fn (x1 ) · · · fn (xn ) i = 1,...,n
Mathematics subject classification numbers: 46B05, 46B20, 46A45, 46A99, 46B99. Key words and phrases: n-normed spaces, p-summable sequence spaces, norm equivalence. 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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R. A. WIBAWA-KUSUMAH and H. GUNAWAN
Here X ′ denotes the dual of X, which consists of bounded For X = ℓp (1 ≤ p < ∞), the space of p-summable bers), the above formula reduces to P x1j z1j · · · ′ .. .. kx1 , . . . , xn kp := sup . . p′ P zi ∈ℓ , kzi kp′ ≤1 x z · · · nj 1j i = 1,...,n
linear functionals on X. sequences (of real num-
P
, P xnj znj x1j znj .. .
′
where k · kp′ denotes the usual norm on X ′ = ℓp and each of the sums is taken over j ∈ N. Here p′ denotes the dual exponent of p, so that 1p + p1′ = 1. In 2001, Gunawan [7] defined a different n-norm on ℓp (1 ≤ p < ∞) by x1j1 1 X X . kx1 , . . . , xn kp := ··· abs .. n! j jn x1j 1 n
p 1/p · · · xnj1 .. .. . . , · · · xnjn
where xi = (xij ), i = 1, . . . , n. For p = 2, this formula reduces to hx1 , x1 i · · · hx1 , xn i 1/2 .. .. .. kx1 , . . . , xn k2 = , . . . hxn , x1 i · · · hxn , xn i where hxi , xj i denotes the usual inner product on ℓ2 . Here kx1 , . . . , xn k2 represents the volume of the n-dimensional parallelepiped spanned by x1 , . . . ,
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