A normed vector space is a real or complex vector space in which a norm has been defined. Formally, one says that a normed vector space is a pair (V, ∥ · ∥) where V is a vector space over K and ∥ · ∥ is a norm in V , but then one usually uses the usual abuse of language and refers to V as being the normed space. Sometimes (frequently?) one has to consider more than one norm at the same time; th...

In this paper, we focuse on different important properties in fuzzy (set-valued) measure theory, such as, non-atomicity, diffusion, regularity, order continuity, exhaustivity, countable subadditivity, increasing/decreasing convergence, semi-convexity, Darboux property for different types of set multifunctions defined on a ring of subsets of an abstract, nonvoid space and taking values in the fa...

Motivated by an Arens regularity problem, we introduce the concepts of matrix Banach space and matrix Banach algebra. The notion of matrix normed space in the sense of Ruan is a special case of our matrix normed system. A matrix Banach algebra is a matrix Banach space with a completely contractive multiplication. We study the structure of matrix Banach spaces and matrix Banach algebras. Then we...

Consider a sum-product normed space, i.e. a space of the form Y = `1 ⊗ X , where X is another normed space. Each element in Y consists of a length-n vector of elements in X , and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constant-distortion embedding from the normed space `1 ⊗X into a lower-dimensional normed space ` ′ 1 ⊗ X , where n′ n is ...

Abstract. Generalizing Pisier’s idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal is the Schatten class. In general, a pair of symmetric normed ideals that are not necessarily the Schatten class may give rise to an operator space...

the main goal of this paper is the study of the generalized hyers-ulam stability of the following functionalequation f (2x  y)  f (2x  y)  (n 1)(n  2)(n  3) f ( y)  2n2 f (x  y)  f (x  y)  6 f (x) where n  1,2,3,4 , in non–archimedean spaces, by using direct and fixed point methods.

In the present paper, we prove that the classical sets ∞(F ), c(F ), c0(F ) and p(F ) of sequences of fuzzy numbers are normed quasilinear spaces and the β−, α−duals of the set 1(F ) is the set ∞(F ). Besides this, we show that ∞(F ) and c(F ) are normed quasialgebras and an operator defined by an infinite matrix belonging to the class ( ∞(F ) : ∞(F )) is bounded and quasilinear. Finally, as an...