Embedding of level-continuous fuzzy sets on Banach spaces

Abstract

Abstract In this paper we present an extension of the Minkowski embedding theorem , showing the existence of an isometric embedding between the class F c ( X ) of compact-convex and level-continuous fuzzy sets on a real separable Banach space X and C ([0,1]×B( X ∗ )) , the Banach space of real continuous functions defined on the cartesian product between [0,1] and the unit ball B( X ∗ ) in the dual space X ∗ . Also, by using this embedding, we give some applications to the characterization of relatively compact subsets of F c ( X ) . In particular, an Ascoli–Arzela type theorem is proved and applied to solving the Cauchy problem x (t)=f(t,x(t)) , x(t0)=x0 on F c ( X ) .