We provide a rule to calculate the subdifferential of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space. Our formula is given exclusively in terms of the data functions, and does not require any assumption either on the index set on which the supremum is taken or on the involved functions. Some other calculus rules, namel...

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...

In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and pre -subbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

We give a lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n − 1) such trajectories. Convex plane billiards were studied by G. Birkhoff, and the case of higher dimensional billiards is considered in our previous papers. We apply a topological approach based on the calculation of cohomol...

We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a complex quasitorus. We associate to each simple polytope, rational or not, a family of complex quasifolds having same dimension as the polytope, each containing a ...

The first point here is that convex sets can be separated by linear functionals. Second, continuous linear functionals on subspaces of a locally convex topological vectorspace have continuous extensions to the whole space. Proofs are for real vectorspaces. The complex versions are corollaries. A crucial corollary is that on locally convex topological vectorspaces continuous linear functionals s...

The Schauder-Tychonoff theorem states that a continuous function from a compact convex subset of a locally convex topological vector space into itself must have a fixed point ([1, Chapter V, 10.5], or [2]). Using this theorem, we obtain here a stronger result, stating that a map from such a set into the surrounding vector space has a fixed point if the directions in which the points are moved s...

If (X, τ) is a topological space, we denote by τs the topology on X whose members are the sets A ⊆ X such that X \A is sequentially τ -closed. Clearly, τs is stronger than τ . We note that a function f : X → R is sequentially τ -lower semicontinuous if and only if it is τs-lower semicontinuous. We also recall that a real function φ on a convex subset C of a vector space is said to be quasi-conc...

In this work, we introduce notion of connectedness on fuzzy soft topological spaces and present fundamentals properties. We also investigate effect to fuzzy soft connectedness. Moreover, $C_i$-connectedness which plays an important role in fuzzy topological space extend to fuzzy soft topological spaces.