In this paper, we formalize the Menger probabilistic normed space as a category in which its objects are the Menger probabilistic normed spaces and its morphisms are fuzzy continuous operators. Then, we show that the category of probabilistic normed spaces is isomorphicly a subcategory of the category of topological vector spaces. So, we can easily apply the results of topological vector spaces...

We use the following convention: m, n denote non empty elements of N, i, j denote elements of N, and Z denotes a set. One can prove the following propositions: (1) Let S, T be real normed spaces, f be a point of the real norm space of bounded linear operators from S into T , and r be a real number. Suppose 0 ≤ r and for every point x of S such that ‖x‖ ≤ 1 holds ‖f(x)‖ ≤ r · ‖x‖. Then ‖f‖ ≤ r. ...

We define the (w∗-) boundedness property and the (w∗-) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called (w∗-) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of tw...

Given ann-normed space withn≥ 2, we offer a simple way to derive an (n−1)norm from the n-norm and realize that any n-normed space is an (n−1)-normed space. We also show that, in certain cases, the (n−1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the n-norm is equivalent to those in the derived (n− 1)-norm. Using this fact, we prove a fixed point t...

Real and complex norms of a linear operator acting on a normed complexified space are considered. Bounds on the ratio of these norms are given. The real and complex norms are shown to coincide for four classes of operators: 1. real linear operators from Lp(μ1) to Lq(μ2), 1 ≤ p ≤ q ≤ ∞; 2. real linear operators between inner product spaces; 3. nonnegative linear operators acting between complexi...

The object of this paper is to introduce the notion of intuitionistic fuzzy continuous mappings and intuitionistic fuzzy bounded linear operators from one intuitionistic fuzzy n-normed linear space to another. Relation between intuitionistic fuzzy continuity and intuitionistic fuzzy bounded linear operators are studied and some interesting results are obtained.

The famous Lomonosov’s invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E “commutes” with a compact operator K 6= 0, i.e., TK = KT, then T has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide some applications to singlevalued linear operators.

the notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $cat(0)$ space, where the curvature is bounded from above by zero. in fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. in this paper, w...

A dilatation structure on a metric space, is a notion in between a group and a differential structure. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. There are two notions of linearity associated to dilatation structures: the linearity of a function between two dilata...

Abstract In this paper, the concept of generalized saddle point(GSP) is employed to discuss the optimization problems of a set of convex functions on a normed linear space X , which presents an equivalence under a special condition between GSP and its optimum solution. A study on integrated convex optimization problem by using Gâteaux and Fréchet differentiability respectivly, and the equivalen...