A new norm on BLO and matters of approximability and duality

TL;DR

This paper introduces a new norm on BLO that makes the distance to L-infinity explicitly computable, explores approximation properties, and establishes an isometric duality between VLO and VMO*.

Contribution

It provides an equivalent norm on BLO with explicit distance computation, proves the norm attaining property, and establishes duality between VLO and VMO*.

Findings

New norm on BLO makes distance to L-infinity explicit and computable.

Functions in BLO can be approximated by truncation and convolution under certain conditions.

Dual of VLO is isometric to VMO*.

Abstract

In this paper, we obtain an alternative expression for the distance of a function in $BLO$ from the subspace $L^\infty$. The distance is the one induced by choosing a new "norm" on $BLO$, equivalent to the usual one and that has the advantage to make the distance to $L^{\infty}$ explicitely and exactly computable. We also prove that this new norm has got the norm attaining property on $BLO$. \\ We address the issue of approximability by truncation as it is not obvious even in the closure of $L^\infty$ in $BLO$ that functions can be approximated by truncation. As a matter of fact, a few examples of this are presented. The easier issue of approximability by convolution is also addressed.\\ In the last section we finally prove that the "dual" of $VLO$ is isometric to $VMO^{\ast}$.