where a∈ R,{bi}i=1 is an arbitrary sequence of real numbers, {τi} ∞ i=1 is a strictly increasing sequence of strictly positive reals such that limi→∞τi = ∞ and φ : (−∞,0] −→ R is continuous. For the special case {bi}i=1 ∈ l 1, (1.1) can be uniquely solved for any given φ ∈ BC (−∞,0], the space of all bounded real-valued continuous functions. The proof of this is indicated in Example1.2. Denote ...

Interest in Sobolev type equations has recently increased signi cantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to ll up the theory but by the aspiration to comprehend nonclassical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if soluti...

Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an ( n + 1)-dimensional multilinear map and -fold composition of linear maps on real normed spaces. result is used to describe space nth-order derivatives Frechet derivative as a space. In Section 1, we discuss 1-dimensional 0-fold compositions preparation, 2, extend discussion compositions...

We initiate a study of non-commutative Choquet boundary for spaces unbounded operators. define the notion local representations operator systems in locally C⁎-algebras and prove that provide an intrinsic invariant particular class systems. An appropriate analog purity completely positive maps on is used to characterize Frechet C⁎-algebras.

Some basic concepts for functions defined on subsets of the unit sphere, such as s-directional derivative, s-gradient and s-Gateaux s-Frechet differentiability etc, are introduced investigated. These different from usual ones Euclidean spaces, however, results obtained here very similar. Then, applications, we provide some criterions s-convexity spheres which improvements or refinements known r...

1. N o t a t i o n a n d pre l iminary ideas . A sequence space is a vector subspace of the space co of all real (or complex) sequences. A sequence space E with a locally convex topology r is called a Kspace if the inclusion map E —* co is continuous, when co is endowed with the product topology (co = II^Li (R)*). A i£-space E with a Frechet (i.e., complete, metrizable and locally convex) topol...

We are to discuss how to view notions of computability for discontinuous functions. We con ne ourselves to real-valued functions from some spaces. Our standpoint in studying computability problems in mathematics is doing mathematics. That is, we would like to talk about computable functions and other mathematical objects just as one talks about continuous functions, integrable functions, etc. I...

It is shown that if E is a Frechet space with the strong dual E∗ then Hb(E ∗), the space of holomorphic functions on E∗ which are bounded on every bounded set in E∗, has the property (DN) when E ∈ (DN) and that Hb(E∗) ∈ (Ω) when E ∈ (Ω) and either E∗ has an absolute basis or E is a Hilbert-Frechet-Montel space. Moreover the complementness of ideals J(V ) consisting of holomorphic functions on E...