Papers by Pradeep Kumar Singh
Singh and Srivastava [9] have studied n-normed structure of ґ by introducing a new n-norm .,.... more Singh and Srivastava [9] have studied n-normed structure of ґ by introducing a new n-norm .,.,…,. on it and have observed that ґ, .,.,…,. is not complete in general. Here we shall investigate some sufficient condition for a Cauchy sequence to be convergent in the n-normed space ґ, .,.,…,. and some new results on Cauchy sequences.
In [4], we have already studied the space of p-summable sequences i.e. (l^
p)
as an n-normed sp... more In [4], we have already studied the space of p-summable sequences i.e. (l^
p)
as an n-normed space by defining a new n-norm on it. In [5], we have resulted
that equivalent norms can be derived by non-equivalent n-norms. Inspired by the
problem raised in Gunawan and others paper [7], in this paper, we shall show
that sequentially equivalent n-norms need not be equivalent.
As, we know that, every findings in Mathematics are valuable for review and reanalysis. Here, we ... more As, we know that, every findings in Mathematics are valuable for review and reanalysis. Here, we reviewed some
n-norms defined on 𝑙^
𝑝 , and introduced a new n-norms on 𝑙^
𝑝 . Which contains some different properties than others.
International Journal of Mathematical Archive, 2016
I n this paper, we shall investigate equivalence relation between n-norms and derived norms defin... more I n this paper, we shall investigate equivalence relation between n-norms and derived norms defined on l^p . Moreover, we shall show that non-equivalent - can also give equivalent
In this paper, we shall investigate equivalence relation between 𝑛-norms and 𝑑erived norms defined on
𝒍^𝒑; 1 ≤ 𝑝 < ∞. Moreover, we shall show that non-equivalent 𝑛-norms can also give equivalent derived norms.
Thesis Chapters by Pradeep Kumar Singh
D.D.U. Gorakhpur University Gorakhpur , 2016
The present thesis entitled “A Study of Topological Linear Structure and Operator Duals of Certai... more The present thesis entitled “A Study of Topological Linear Structure and Operator Duals of Certain Vector Valued Sequence Spaces” is the outcome of the investigations carried out by Mr. Pradeep Kumar Singh towards the fulfillment for the award of degree of the Doctor of Philosophy in Mathematics under the supervision of Prof. J K Srivastava, Department of Mathematics and Statistics, D D U Gorakhpur University, Gorakhpur. We know that sequence space play a vital role in Functional Analysis. In this direction our aim is to introduce and study certain new n-norm structure of sequence space l^p and equivalent relations of different n-norms together with the n-dual structure of sequence space l^p. The Thesis has been divided into five chapters with first as Introduction. Further, each chapter is subdivided into several sections and subsections. At the end of the chapter 5, bibliography has been attached and after that a list of published/communicated papers has been enclosed. At last, 2 out of 5 published papers have been enclosed. Chapter 1 is introductory in nature and has been subdivided into two sections namely, PART-A and PART-B. In PART-A a brief survey about the notion of n-normed space is given. While in PART-B, some well-known definitions and results of Functional Analysis connected to our work have been given with citation from some standard text books or monographs for ready references. In Chapter 2, we have introduced a new n-norm, ‖.,…,.‖ ����������� on l^p and a new notion of parallel rearranged sequences. We have investigated the relations between new n-norm and wellknown norms on l^p. Further, we have investigated the various aspects of analysis connected to Cauchy sequences in new n-norm sequence space �ѝ,‖.,…,.‖Chapter 3 deals with topological linear structure of n-norm space �ѝ,‖.,…,.‖The motivation of this work is that- Can we derive equivalent norms by non-equivalent n-norms? Can non-equivalent n-norms be sequentially equivalent? In this chapter, we have investigated the relations between derived norms, n-norms and various aspects of equivalence and sequentially equivalence relations of derived norms and n-norms. Chapter 4 deals with n-dual structure of sequence space ѝ with respect to various n-norms defined by us and others. We also deal with containment relations of different n-dual spaces of ѝ when ѝ is equipped with various n-norms. The motivation of the work of Chapter 5 is that- Can we extend the domain of n-norm ‖.,…,.‖ �����������from ѝ to other sequence spaces? In this chapter, we have investigated results related to ‖.,…,.‖ �����������and have constructed some new sequence space which lie between ѝ and Ȟ.
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Papers by Pradeep Kumar Singh
p)
as an n-normed space by defining a new n-norm on it. In [5], we have resulted
that equivalent norms can be derived by non-equivalent n-norms. Inspired by the
problem raised in Gunawan and others paper [7], in this paper, we shall show
that sequentially equivalent n-norms need not be equivalent.
n-norms defined on 𝑙^
𝑝 , and introduced a new n-norms on 𝑙^
𝑝 . Which contains some different properties than others.
In this paper, we shall investigate equivalence relation between 𝑛-norms and 𝑑erived norms defined on
𝒍^𝒑; 1 ≤ 𝑝 < ∞. Moreover, we shall show that non-equivalent 𝑛-norms can also give equivalent derived norms.
Thesis Chapters by Pradeep Kumar Singh
p)
as an n-normed space by defining a new n-norm on it. In [5], we have resulted
that equivalent norms can be derived by non-equivalent n-norms. Inspired by the
problem raised in Gunawan and others paper [7], in this paper, we shall show
that sequentially equivalent n-norms need not be equivalent.
n-norms defined on 𝑙^
𝑝 , and introduced a new n-norms on 𝑙^
𝑝 . Which contains some different properties than others.
In this paper, we shall investigate equivalence relation between 𝑛-norms and 𝑑erived norms defined on
𝒍^𝒑; 1 ≤ 𝑝 < ∞. Moreover, we shall show that non-equivalent 𝑛-norms can also give equivalent derived norms.