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Mappings of type generalized de La Vallée Poussin’s mean

Awad A Bakery

Author Affiliations

Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais, Code 21589, Saudi Arabia

Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, P.O. Box 1156, Cairo, 11566, Egypt

Journal of Inequalities and Applications 2013, 2013:518  doi:10.1186/1029-242X-2013-518

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/518


Received:21 April 2013
Accepted:9 September 2013
Published:9 November 2013

© 2013 Bakery; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the present paper, we study the operator ideals generated by the approximation numbers and generalized de La Vallée Poussin’s mean defined in (Şimşek et al. in J. Comput. Anal. Appl. 12(4):768-779, 2010). Our results coincide with those in (Faried and Bakery in J. Inequal. Appl. 2013, doi:10.1186/1029-242X-2013-186) for the generalized Cesáro sequence space.

Keywords:
approximation numbers; operator ideal; generalized de La Vallée Poussin’s mean sequence space

1 Introduction

By <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M1">View MathML</a> we denote the space of all bounded linear operators from a normed space X into a normed space Y. The set of nonnegative integers is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M2">View MathML</a> and the real numbers by ℝ. By ω we denote the space of all real sequences. A map which assigns to every operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3">View MathML</a> a unique sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M4">View MathML</a> is called an s-function and the number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M5">View MathML</a> is called the nth s-numbers of T if the following conditions are satisfied:

(a) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M6">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3">View MathML</a>.

(b) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M8">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M9">View MathML</a>.

(c) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M10">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M11">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M12">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M13">View MathML</a>.

(d) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M14">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M16">View MathML</a>.

(e) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M17">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3">View MathML</a>.

(f)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M21">View MathML</a> is the identity operator on the Euclidean space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M22">View MathML</a>.

As examples of s-numbers, we mention approximation numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M23">View MathML</a>, Gelfand numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M24">View MathML</a>, Kolmogorov numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M25">View MathML</a> and Tichomirov numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M26">View MathML</a> defined by:

(I) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M27">View MathML</a>.

(II) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M28">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M29">View MathML</a> is a metric injection (a metric injection is a one-to-one operator with closed range and with norm equal to one) from the space Y into a higher space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M30">View MathML</a> for a suitable index set Λ.

(III) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M31">View MathML</a>.

(IV) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M32">View MathML</a>.

All these numbers satisfy the following condition:

(g) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M33">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M34">View MathML</a>.

An operator ideal U is a subclass of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M35">View MathML</a> such that its components <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M36">View MathML</a> satisfy the following conditions:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M37">View MathML</a>, where K denotes the 1-dimensional Banach space, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M38">View MathML</a>.

(ii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M39">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M40">View MathML</a> for any scalars <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M41">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M42">View MathML</a>.

(iii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M43">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M44">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M13">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M46">View MathML</a>. See [1,2] and [3].

For a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M47">View MathML</a> of positive real numbers with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M48">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49">View MathML</a>, the generalized Cesáro sequence space is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M50">View MathML</a>

The space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M51">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M52">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M47">View MathML</a> is bounded, we can simply write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M54">View MathML</a>. Also, some geometric properties of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M51">View MathML</a> are studied in [4-6] and [7].

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M56">View MathML</a> be a nondecreasing sequence of positive real numbers tending to infinity, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M57">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M58">View MathML</a>.

De La Vallée Poussin’s means of a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M59">View MathML</a> are defined as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M60">View MathML</a>

The generalized de La Vallée Poussin’s mean sequence space was defined in [8].

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M61">View MathML</a>

The space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M63">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M47">View MathML</a> is bounded, we can simply write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M65">View MathML</a>

Also, some geometric properties of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a> are studied in [9,10] and [11].

Throughout this paper, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a> is a bounded sequence of positive real numbers with

(b1) the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a> of positive real numbers is increasing and bounded with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M69">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M70">View MathML</a>,

(b2) the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M71">View MathML</a> is a nondecreasing sequence of positive real numbers tending to infinity, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M57">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M73">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M74">View MathML</a>.

Also we define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M75">View MathML</a>, where 1 appears at the ith place for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M76">View MathML</a>.

Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [12-15] and [16].

For any bounded sequence of positive numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a>, we have the following well-known inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M78">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M79">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M48">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M81">View MathML</a>. See [17].

2 Preliminary and notation

Definition 2.1 A class of linear sequence spaces E is called a special space of sequences (sss) having the following conditions:

(1) E is a linear space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M82">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49">View MathML</a>.

(2) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M84">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M85">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M86">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M88">View MathML</a>i.e., E is solid’.

(3) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M89">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M90">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M91">View MathML</a> denotes the integral part of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M92">View MathML</a>.

Example 2.2<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M93">View MathML</a> is a special space of sequences for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M94">View MathML</a>.

Example 2.3<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M95">View MathML</a> is a special space of sequences for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M96">View MathML</a>.

Definition 2.4<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M97">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M98">View MathML</a>.

Theorem 2.5<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M99">View MathML</a>is an operator ideal ifEis a special space of sequences (sss).

Proof See [18]. □

We give here the sufficient conditions on the generalized de La Vallée Poussin’s mean such that the class of all bounded linear operators between any arbitrary Banach spaces with nth approximation numbers of the bounded linear operators in the generalized de La Vallée Poussin’s mean form an operator ideal.

3 Main results

Theorem 3.1<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M100">View MathML</a>is an operator ideal, if conditions (b1) and (b2) are satisfied.

Proof (1-i) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M101">View MathML</a> since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M102">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M104">View MathML</a>.

(1-ii) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M16">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M106">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M107">View MathML</a>

we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M108">View MathML</a>, from (1-i) and (1-ii), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a> is a linear space.

To show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M110">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M111">View MathML</a>, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M74">View MathML</a>. Thus we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M113">View MathML</a>

Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M110">View MathML</a>.

(2) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M115">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M117">View MathML</a> since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M118">View MathML</a>. Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M119">View MathML</a>.

(3) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M120">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M121">View MathML</a>

Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M122">View MathML</a>. Hence from Theorem 2.5 it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M100">View MathML</a> is an operator ideal. □

Corollary 3.2<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M124">View MathML</a>is an operator ideal if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a>is an increasing sequence of positive real numbers, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M126">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M127">View MathML</a>.

Corollary 3.3<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M128">View MathML</a>is an operator ideal if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M96">View MathML</a>.

Theorem 3.4The linear space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M130">View MathML</a>is dense in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M131">View MathML</a>if conditions (b1) and (b2) are satisfied.

Proof First we prove that every finite mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M132">View MathML</a> belongs to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M131">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M110">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M135">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a> is a linear space, then for every finite mapping <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M132">View MathML</a>, i.e., the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M138">View MathML</a> contains only finitely many numbers different from zero. Now we prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M139">View MathML</a>. Since letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M140">View MathML</a> we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M141">View MathML</a>, and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M142">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M143">View MathML</a>, then there exists a natural number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M144">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M145">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M146">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M147">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M148">View MathML</a> is decreasing for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M49">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M150">View MathML</a>

(1)

then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M151">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M152">View MathML</a> with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M153">View MathML</a>

(2)

and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a> is a bounded sequence of positive real numbers, so we can take

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M155">View MathML</a>

(3)

also <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M156">View MathML</a>. Then there exists a natural number <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M158">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M159">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M160">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M161">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M162">View MathML</a>

(4)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a> is an increasing sequence, by using (1), (2), (3) and (4), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M164">View MathML</a>

 □

Definition 3.5 A class of special space of sequences (sss) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M165">View MathML</a> is called a pre-modular special space of sequences if there exists a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M166">View MathML</a> satisfying the following conditions:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M167">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M168">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M169">View MathML</a>, where θ is the zero element of E,

(ii) there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M171">View MathML</a> for all values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M88">View MathML</a> and for any scalar λ,

(iii) for some numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M173">View MathML</a>, we have the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M174">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M175">View MathML</a>,

(iv) if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M81">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M178">View MathML</a>,

(v) for some numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M179">View MathML</a>, we have the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M180">View MathML</a>,

(vi) for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M181">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M182">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M183">View MathML</a>. This means the set of all finite sequences is ρ-dense in E,

(vii) for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M184">View MathML</a>, there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M185">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M186">View MathML</a>.

It is clear from condition (ii) that ρ is continuous at θ. The function ρ defines a metrizable topology in E endowed with this topology which is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M165">View MathML</a>.

Example 3.6<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M93">View MathML</a> is a pre-modular special space of sequences for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M189">View MathML</a>, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M190">View MathML</a>.

Example 3.7<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M95">View MathML</a> is a pre-modular special space of sequences for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M192">View MathML</a>, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M193">View MathML</a>.

Theorem 3.8<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M195">View MathML</a>is a pre-modular special space of sequences if conditions (b1) and (b2) are satisfied.

Proof (i) Clearly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M167">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M197">View MathML</a>.

(ii) Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a> is bounded, then there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M171">View MathML</a> for all values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M88">View MathML</a> and for any scalar λ.

(iii) For some numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M202">View MathML</a>, we have the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M203">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M101">View MathML</a>.

(iv) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M206">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M207">View MathML</a>.

(v) There exist some numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M208">View MathML</a>; by using (iv) we have the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M209">View MathML</a>.

(vi) It is clear that the set of all finite sequences is ρ-dense in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a>.

(vii) For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M184">View MathML</a>, there exists a constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M212">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M213">View MathML</a>. □

Theorem 3.9LetXbe a normed space, Ybe a Banach space, and let conditions (b1) and (b2) be satisfied, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M214">View MathML</a>is complete.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M215">View MathML</a> be a Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M214">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M62">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M218">View MathML</a> is a pre-modular special space of sequences, then, by using condition (vii) and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M219">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M220">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M215">View MathML</a> is also a Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M1">View MathML</a>. Since the space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M1">View MathML</a> is a Banach space, then there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M3">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M225">View MathML</a> and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M226">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M227">View MathML</a>, ρ is continuous at θ and using (iii), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M228">View MathML</a>

Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M229">View MathML</a> as such <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M230">View MathML</a>. □

Corollary 3.10LetXbe a normed space, Ybe a Banach space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a>be an increasing sequence of positive real numbers with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M232">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M233">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M234">View MathML</a>is complete.

Corollary 3.11LetXbe a normed space, Ybe a Banach space and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M67">View MathML</a>be an increasing sequence of positive real numbers with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M96">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/518/mathml/M237">View MathML</a>is complete.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author is most grateful to the editor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper.

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