F-seminorms on generalized double sequence spaces defined by modulus functions

A Raidjõe,E Kolk

doi:10.3176/proc.2014.2.02

Abstract

Using a double sequence of modulus functions and a solid double scalar sequence space, we determine F-seminorm and F-norm topologies for certain generalized linear spaces of double sequences. The main results are applied to the topologization of double sequence spaces related to 4-dimensional matrix methods of summability.

Highlights

Let N = {1, 2, . . . } and let K be the field of real numbers R or complex numbers C
The main results are applied to the topologization of double sequence spaces related to 4-dimensional matrix methods of summability
In the following we extend these results to the generalized double sequence spaces defined by means oanf oathlienredaor uobpleerasteoqruTen:cse2To(fXs2e)m→inosr2m(Ye2d)swpaitchesT(xY2ki=, ̇|

Summary

INTRODUCTION

Let X2 be a double sequence of seminormed linear spaces Xki, ̇| · ̇|ki (k, i ∈ N). Any linear subspace of s2(X2) is called a generalized double sequence space (GDS space). For example, s2 denotes the linear space of all K-valued double sequences u2 = (uki). GDS space Λ(X2) ⊂ s2(X2) is (k, i ∈ N) It is not called solid if difficult to see (yki) ∈ Λ(X2) that the sets whenever (xki). Λ Φ, X2 = x2 ∈ s2(X2) : Φ(x2) = φki|xki|ki ∈ Λ , and Λ(Φ, X2)∩s2(X2) are solid GDS spaces if Λ ⊂ s2 is a solid DS space and Φ = (φki) is a double sequence of moduli. Some special cases of such spaces are considered, for example, in [1,3,4,15,17]

MAIN THEOREMS

SOME APPLICATIONS

CONCLUSION