EQUIVALENCE OF n-NORMS ON THE SPACE OF p-SUMMABLE SEQUENCES (original) (raw)

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.

Two equivalent n-norms on the space of p-summable sequences

Periodica Mathematica Hungarica, 2013

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 intoduced by White (1969), and studied further by Batkunde et al., and Gozali et al. (2010). 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.

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.

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].

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 .

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.

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 .

A Note on Generalized Holder’s Inequality in P-summable Sequence Spaces

2020

In this paper, we present the sufficient and necessary conditions for generalized H?̈?lder’s inequality in weak p-summable sequence spaces which complete the Masta, et al. results in 2018. One of the keys to prove our results is to use the norm of the characteristic sequence of the balls in Z.

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.

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.