Some properties of Diametric norm and its isometries on B_0 (Bbb R)$

Hagler used a diameter norm to construct a separable Banach space 𝑋 with nonseparable dual such that 𝑙1 does not embed in 𝑋. Also, Bayati Eshkaftaki considered the diameter norm on 𝑐0(𝐼) and characterized all isometries on this space. In this article, we are going to first introduce the space 𝐵0 (ℝ) and show that this space is Banakh space with supremum norm, and then we express the diameter norm on the space 𝐵0(ℝ) and examine some of the characteristics of this space. We also examine the relationship between this space and Banach space 𝑐0(ℝ) and some of the reference theorems [3] shown on space 𝑐0(ℝ)on space 𝐵0(ℝ).