A new norm on BLO and matters of approximability and duality

In this paper, we obtain an alternative expression for the distance of a function in $BLO$ from the subspace $L^{\mathrm{\infty }}$. 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 $L^{\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$ 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 $VM{O}^{\ast }$.