We give a full characterization of normability of Lorentz spaces ${\Gamma}_{w}^{p}$. This result is in fact known since it can be derived from Kamińska A., Maligranda L., On Lorentz spaces, Israel J. Funct. Anal. 140 (2004), 285–318. In this paper we present an alternative and more direct proof.

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...

We characterize associate spaces of weighted Lorentz spaces GΓ(p,m,w) and present some applications of this result including necessary and sufficient conditions for a Sobolev-type embedding into ${L}^{\infty}$.

The weak lower semicontinuity of the functional $$F\left(u\right)={\int}_{\Omega}f(x,u,\nabla u)\mathrm{d}x$$
is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on ${W}^{1,p}\left(\Omega \right)$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.

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