Normability of Lorentz spaces—an alternative approach

summary:We study normability properties of classical Lorentz spaces. Given a certain general lattice-like structure, we first prove a general sufficient condition for its associate space to be a Banach function space. We use this result to develop an alternative approach to Sawyer's characterization of normability of a classical Lorentz space of type $\Lambda $. Furthermore, we also use this method in the weak case and characterize normability of $\Lambda _{v}^{\infty }$. Finally, we characterize the linearity of the space $\Lambda _{v}^{\infty }$ by a simple condition on the weight $v$