This defines an operator MK̄ , which we will call a Minkowski operator. Denote by A(C) the Frechet space of entire functions in n variables with the usual topology of the uniform convergence on compact sets in C, and C(R) the Frechet space of r times differentiable functions on R with the topology of the uniform convergence on compact sets in R of all partial derivatives up to the order r (1 ≤ r...

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to...

and Applied Analysis 3 2. Preliminary Results on Closed and Bounded Sequences of Linear Operators The subsequent result relies on the convergence of sequences of operators to limits so that we give the following simple result. Theorem 2.1. P∗1 / P∗2 ⇒ S P∗1 / S P∗2 ; for all P∗1, P∗2 ∈ L X ∩ clFP . Proof. Pk ∈ S P∗1 ⇔ Pk → P∗1 as k → ∞ in the strong operator topology. Thus, for any ε1 > 0, ∃n1 ...

Several recent publications have exhibited relationships between the theories of logic programming and of neural networks. We consider a general approach to representing normal logic programs via feedforward neural networks. We show that the immediate consequence operator associated with each logic program, which can be understood as implicitly determining its declarative semantics, can be appr...

We present an operator space version of Rieffel’s theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As an application we obtain a necessary and sufficient condition for the matrix metric from an unbounded Fredholm module to give the BW-topology on the matrix state space of the C-algebra. Motivated by...

The concept of an M-approximative system is introduced. Basic properties of the category of M-approximative systems and in a natural way defined morphisms between them are studied. It is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the category of M-approximative systems. Keywords— Approximative operator...

Equilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm...

The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in R, and on the half space R + . The novelty is that the topology used is not an L 2 topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary value problem belonging to a class of problems first intr...