In this paper, a complete description concerning linear operators of Banach spaces with range in Lipschitz algebras $lip_al(X)$ is provided. Necessary and sufficient conditions are established to ensure boundedness and (weak) compactness of these operators. Finally, a lower bound for the essential norm of such operators is obtained.

In [3] and [6] the concepts of smooth closure, smooth interior, smooth α-closure and smooth α-interior of a fuzzy set were introduced and some of their properties were obtained. In this paper, we introduce the concepts of several types of weak smooth compactness and weak smooth α-compactness in terms of these concepts introduced in [3] and [6] and investigate some of their properties.

• Banach-Alaoglu: compactness of polars • Variant Banach-Steinhaus/uniform boundedness • Second polars • Weak boundedness implies boundedness • Weak-to-strong differentiability The comparison of weak and strong differentiability is due to Grothendieck, although the original sources are not widely available.

Abstract. The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a d...