Fundamental Properties of Absolute-Type Cesàro Sequence Space on Metric Spaces

The continuing development and interest of mathematical researchers, both from the purely theoretical and the applied perspective, can still be seen in the field of sequence spaces.One such sequence space is the Cesàro sequence of an absolute space, which J. S. Shiue introduced for real sequences in 1970.Let 𝟏𝟏 ≤ 𝒑𝒑 < ∞ be real numbers.This space is defined to be the set of all real sequences (𝒙𝒙 𝒏𝒏 ) such that ∑ � 𝟏𝟏 𝒏𝒏 ∑ |𝒙𝒙 𝒌𝒌 | 𝒏𝒏 𝒌𝒌=𝟏𝟏 � 𝒑𝒑 ∞ 𝒏𝒏=𝟏𝟏 < ∞.In ℝ, the terms |𝒙𝒙 𝒌𝒌 |, relates to the standard metric 𝒅𝒅 ℝ (𝒂𝒂, 𝒃𝒃) ≔ |𝒂𝒂 -𝒃𝒃|, ∀𝒂𝒂, 𝒃𝒃 ∈ ℝ, thus |𝒙𝒙 𝒌𝒌 | = 𝒅𝒅 ℝ (𝒙𝒙 𝒌𝒌 , 𝟎𝟎).Aiming to explore this insight, we construct the definition of absolute Cesàro sequence space on an arbitrary metric space (𝑿𝑿, 𝒅𝒅) with a fixed point 𝒙𝒙 𝟎𝟎 ∈ 𝑿𝑿, denoted by 𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ).Then, we define a metric on 𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), denoted by 𝝆𝝆 𝒑𝒑 .Also, we investigate the effect of the choice of 𝒙𝒙 𝟎𝟎 ∈ 𝑿𝑿 and the fundamental aspects of �𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), 𝝆𝝆 𝒑𝒑 � such as completeness, the inclusion relation of 𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), the ordering of 𝝆𝝆 𝒑𝒑 on elements of 𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ) for various 𝒑𝒑, and the separability of �𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), 𝝆𝝆 𝒑𝒑 �.The primary outcomes indicate that for 𝟏𝟏 ≤ 𝒑𝒑 < ∞, �𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), 𝝆𝝆 𝒑𝒑 � is complete if (𝑿𝑿, 𝒅𝒅) is complete; for 𝟏𝟏 < 𝒑𝒑 < ∞, �𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), 𝝆𝝆 𝒑𝒑 � is separable if and only if (𝑿𝑿, 𝒅𝒅) is separable; and �𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ), 𝝆𝝆 𝒑𝒑 � is not complete if (𝑿𝑿, 𝒅𝒅) is not complete.Also, for 𝟏𝟏 ≤ 𝒑𝒑 < 𝒒𝒒 < ∞, 𝒄𝒄𝒄𝒄𝒔𝒔 𝒑𝒑 (𝒙𝒙 𝟎𝟎 ) ⊆ 𝒄𝒄𝒄𝒄𝒔𝒔 𝒒𝒒 (𝒙𝒙 𝟎𝟎 ).For 𝒑𝒑 = 𝟏𝟏, we have 𝒄𝒄𝒄𝒄𝒔𝒔 𝟏𝟏 (𝒙𝒙 𝟎𝟎 ) = {(𝒙𝒙 𝟎𝟎 , 𝒙𝒙 𝟎𝟎 , … )}.Hence, (𝒄𝒄𝒄𝒄𝒔𝒔 𝟏𝟏 (𝒙𝒙 𝟎𝟎 ), 𝝆𝝆 𝟏𝟏 ) is separable for any metric space (𝑿𝑿, 𝒅𝒅).