Two equivalent n-norms on the space of p-summable sequences (original) (raw)

The n-dual space of the space of p-summable sequences

Mathematica Bohemica

In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an n-normed space, we are interested in bounded multilinear n-functionals and n-dual spaces. The concept of bounded multilinear n-functionals on an n-normed space was initially introduced by A. G. White jun. [Math. Nachr. 42, 43–60 (1969; Zbl 0185.20003)] and studied further by H. Batkunde, H. Gunawan and Y. E. P. Pangalela [“Bounded linear functionals on the n-normed space of p-summable sequences”, Acta Univ. M. Belii, Ser. Math. 21, 66–75 (2013), http://actamath.savbb.sk/pdf/aumb2107.pdf\] and S. M. Gozali et al. [Ann. Funct. Anal. AFA 1, No. 1, 72–79 (2010; Zbl 1208.46006)]. In this paper, we revisit the definition of bounded multilinear n-functionals, introduce the concept of n-dual spaces, and then determine the n-dual spaces of ℓ p spaces when these spaces are not only equipped with the usual norm, but also with some n-norms.

Bounded linear functionals on the n-normed space of p-summable sequences

Let (X, ·, · · · , · ) be a real n-normed space, as introduced by S. Gähler in 1969. We shall be interested in bounded linear functionals on X, using the n-norm as our main tool. We study the duality properties and show that the space X of bounded linear functionals on X also forms an n-normed space. We shall present more results on bounded multilinear n-functionals on the space of p-summable sequences being equipped with an n-norm. Open problems are also posed.

Three Equivalent nnn-Norms on the Space of ppp-Summable Sequences

Fundamental Journal of Mathematics and Applications, 2019

Given a normed space, one can define a new n-norm using a semi-inner product g on the space, different from the n-norm defined by Gähler. In this paper, we are interested in the new n-norm which is defined using such a functional g on the space p of p-summable sequences, where 1 ≤ p < ∞. We prove particularly that the new n-norm is equivalent with the one defined previously by Gunawan on p. Then, one may check that g satisfies the following properties: (1) g(x, x) = x 2 for every x ∈ X; (2) g(αx, β y) = αβ g(x, y) for every x, y ∈ X and α, β ∈ R; (3) g(x, x + y) = x 2 + g(x, y) for every x, y ∈ X; (4) |g(x, y)| ≤ x y for every x, y ∈ X. Assuming that the g-functional is linear in the second argument then [y, x] = g(x, y) is a semi-inner product on X. Note that all vector spaces in text are assumed to be over R. For example, one may observe that the functional g(x, y) := x 2−p p ∑ k |x k | p−1 sgn (x k) y k , x := (x k) , y := (y k) ∈ p is a semi-inner product on p , 1 ≤ p < ∞ [1]. Remark 1.1. Note that not all vector spaces have the property that the g-functional is linear in the second argument. If the normed space is smooth, then the g-functional is linear in the second argument. A normed spaces with the property that the g-functional is linear in the second argument is referred to as normed spaces of (G)-type [2].

The space of p-summable sequences and its natural n-norm

Bulletin of The Australian Mathematical Society, 2001

We study the space l p , 1 ≤ p ≤ ∞, and its natural n-norm, which can be viewed as a generalization of its usual norm. Using a derived norm equivalent to its usual norm, we show that l p is complete with respect to its natural n-norm. In addition, we also prove a fixed point theorem for l p as an n-normed space.

EQUIVALENCE OF n-NORMS ON THE SPACE OF p-SUMMABLE SEQUENCES

Journal of the Indonesian Mathematical Society, 2010

We study the relation between two known n-norms on p , the space of p-summable sequences. One n-norm is derived from Gähler's formula [3], while the other is due to Gunawan [6]. We show in particular that the convergence in one n-norm implies that in the other. The key is to show that the convergence in each of these n-norms is equivalent to that in the usual norm on p .

Some notes on the space of p-summable sequences

kuwait journal of science, 2018

Recently, Konca et al. (2015a) have revisited the space p of p -summable sequences of real numbers and have shown that this space is actually contained in a weighted inner product space called 2 v . In another paper, Konca et al. (2015b) have investigated the space p to show that it is also contained in a weighted 2-inner product space. In this work, we show in the details that p is dense in 2 v as a normed space, and as a 2-normed space. Further, we prove that 2 v is separable and conclude that it is isometric to the completion of p .

A new 2-inner product on the space of p -summable sequences

Journal of the Egyptian Mathematical Society, 2016

In this paper, we wish to define a 2-inner product, non-standard, possibly with weights, on p. For this purpose, we aim to obtain a different 2-norm .,. 2, v , w , which is not equivalent to the usual 2-norm .,. p on p (except with the condition p = 2), satisfying the parallelogram law. We discuss the properties of the induced 2-norm .,. 2, v , w and its relationships with the usual 2-norm on p. We also find that the 2-inner product ., .|. v , w is actually defined on a larger space.

A CONTRACTIVE MAPPING THEOREM ON THE n-NORMED SPACE OF p-SUMMABLE SEQUENCES

We revisit the space ℓ of-summable sequences of real numbers which is equipped with an-norm. We derive a norm from the-norm in a certain way, and show that this norm is equivalent to the usual norm on ℓ. We then use this fact together with others to prove a contractive mapping theorem on the-normed space ℓ .

EQUIVALENCE AMONG THREE 2-NORMS ON THE SPACE OF p-SUMMABLE SEQUENCES

2016

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 Gähler’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.

ON A CLASS OF n-NORMED SEQUENCES RELATED TO ellp\ell_pellp-SPACE

Boletim da Sociedade Paranaense de Matemática, 2013

In this paper we introduce the class of n-normed sequences related to the p-absolutely summable sequence space. Some properties of this sequence space like solidness, symmetricity, convergence-free etc. are studied. We obtain some inclusion relations involving this sequence space.