Geodesic
Normability of Lorentz spaces—an alternative approach
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 581-597
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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 $v$.
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 $v$.
DOI : 10.1007/s10587-014-0120-y
Classification : 46E30
Keywords: weighted Lorentz space; weighted inequality; non-increasing rearrangement; Banach function space; associate space 
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Gogatishvili, Amiran; Soudský, Filip. Normability of Lorentz spaces—an alternative approach. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 581-597. doi: 10.1007/s10587-014-0120-y

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