A Note on Generalized Holder’s Inequality in P-summable Sequence Spaces (original) (raw)

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.

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 .

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 .

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.

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.

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.

On p-summable sequences in locally convex spaces

We follow the notation of [7,10]. In particular, if E[?] is a locally convex space (in short l. c. s. ), ?(E; E,); 1(E; E,) and 炉(E; E,) will denote, respecti-vely, the weak, Mackey and strong topology corresponding to the dual pair hE; E,i. If U is a neighbourhood of 0 in E[?], EU denotes the Banach space associated to U and 谩U : E ! EU denotes the corresponding quotient map.

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 .

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].