1.1. Normed spaces. Recall that a (real) vector space V is called a normed space if there exists a function ‖ · ‖ : V → R such that (1) ‖f‖ ≥ 0 for all f ∈ V and ‖f‖ = 0 if and only if f = 0. (2) ‖af‖ = |a| ‖f‖ for all f ∈ V and all scalars a. (3) (Triangle inequality) ‖f + g‖ ≤ ‖f ||+ ‖g‖ for all f, g ∈ V . If V is a normed space, then d(f, g) = ‖f−g‖ defines a metric on V . Convergence w.r.t ...

Let I = [a, b] and let X be a normed space. A function f : I → X is said to be regulated if for all t ∈ [a, b) the limit lims→t+ f(s) exists and for all t ∈ (a, b] the limit lims→t− f(s) exists. We denote these limits respectively by f(t ) and f(t−). We define R(I,X) to be the set of regulated functions I → X. It is apparent that R(I,X) is a vector space. One checks that a regulated function is...

Researchers have identified and defined β- approach normed space if some conditions are satisfied. In this work, we show that every is a space.However, the converse not necessarily true by giving an example. addition, define β – Banach space, examples given. We also solve problems. discuss finite β-dimensional app-normed β-complete consequent app- space. explain metric but give propositions. If...

In this paper, we shall define and study the concept of  -statistical convergence and  -statistical Cauchy in random 2-normed space. We also introduce the concept of  -statistical completeness which would provide a more general frame work to study the completeness in random 2-normed space. Furthermore, we also prove some new results.

in the present paper, we study some properties of fuzzy norm of linear operators. at first the bounded inverse theorem on fuzzy normed linear spaces is investigated. then, we prove hahn banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. finally the set of all compact operators on these spaces is studied.

in this paper, we prove the generalized hyers--ulamstability of a quadratic and quartic functional equation inintuitionistic fuzzy banach spaces.

and Applied Analysis 3 a vector space from various points of view 28–30 . In particular, Bag and Samanta 31 , following Cheng and Mordeson 32 , gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type 33 . They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed ...

We study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesX_{i}. We introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by X boxtimesY. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...

Let X be a non empty complex normed space structure and let s1 be a sequence of X. The functor ( ∑ κ α=0(s1)(α))κ∈N yielding a sequence of X is defined as follows: (Def. 1) ( ∑ κ α=0(s1)(α))κ∈N(0) = s1(0) and for every natural number n holds ( ∑ κ α=0(s1)(α))κ∈N(n + 1) = ( ∑ κ α=0(s1)(α))κ∈N(n) + s1(n + 1). One can prove the following proposition (1) Let X be an add-associative right zeroed rig...

Marouf (1993) presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. Patterson (2003), extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for non-negative sum ability matrices. In Patterson and Savas (2006) extended the definitions presented in (Patterson, 2003) to l...