Geometric Properties of Relative Uniform Lacunary Convergence of Sequences of Functions

Abstract

This paper introduces the concept of relative uniform lacunary convergence for sequences of real
or complex-valued functions. The notion is developed by comparing a given sequence to a reference sequence
under a lacunary summability framework. We explore basic properties such as linearity and completeness
and define classes of relatively lacunary convergent and null sequences. Various topological characteristics of
the function space under this mode of convergence, such as convexity, separability, and symmetry, are also
investigated. Several inclusion results are established and illustrative counterexamples are provided.