Relative uniform convergence of sequence of functions related to $\ell_p$- space

Abstract

The concept of relative uniform convergence of sequence of Sargent type functions spaces $m(\phi,ru)$ and $n(\phi,ru)$ are defined in this article. Here we established a bounded, $p$-absolutely summable, convergent, null sequence of functions using the concept of relative uniform convergent for the scale function and these are denoted by $\ell_{\infty}(ru), \ell_p(ru), c(ru), c_0(ru)$ respectively. We established a relationship between the new notion  $m(\phi,ru)$, $n(\phi,ru)$, and other known concepts. Also, we established some algebraic and geometrical characteristics. Many geometric concepts and difficulties may be solved through an understanding of the characteristics of these special spaces.