Baire-Type Properties in Metrizable c0(Ω, X)

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

Rosenthal L∞-theorem revisited

A remarkable Rosenthal L∞-theorem is extended to operators T : L∞(Γ, E) → F , where Γ is an infinite set, E a locally bounded (for instance, normed or p-normed) space, and F any topological vector space.

A New Notion of g-Angle in Normed

We discuss the g-orthogonality (⊥ g ) and its basic properties in a normed space. We also define a new g-angle between two vectors and prove its basic properties by using the g-orthogonality (⊥ g ). Moreover, we prove the sufficient and necessary conditions for strictly convexity in a normed space by using a new g-angle.

Sequence spaces defined via Euler method and matrix transformations

A blend of matrix summability and Euler summability transformation methods is used to define Lacunary sequence spaces defined over n-normed space. Then we present the properties of this space and finally, some inclusion relations are presented.

on ideal convergence of triple sequences in intuitionistic Fuzzy normed space defined by compact operator

The main purpose of this article is to introduce and study some new spaces of I-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e 3SI (μ,ν)(T ) and 3SI0(μ,ν)(T ) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.

Isometries of ultrametric normed spaces

We show that the group of isometries of an ultrametric normed space can be seen as a kind of a fractal. Then, we apply this description to study ultrametric counterparts of some classical problems in Archimedean analysis, such as the so called Problème des rotations de Mazur or Tingley’s problem. In particular, it turns out that, in contrast with the case of real normed spaces, isometries between ultrametric normed spaces can be very far from being linear.

Applications of Orlicz–Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators

In this study, we deal with some new vector valued multiplier spaces $S_{G_{h}}(\sum_{k}z_{k})$ S G h ( ∑ k z k ) and $S_{wG_{h}}(\sum_{k}z_{k})$ S w G h ( ∑ k z k ) related with $\sum_{k}z_{k}$ ∑ k z k in a normed space Y. Further, we obtain the completeness of these spaces via weakly unconditionally Cauchy series in Y and $Y^{*}$ Y ∗ . Moreover, we show that if $\sum_{k}z_{k}$ ∑ k z k is unconditionally Cauchy in Y, then the multiplier spaces of $G_{h}$ G h -almost convergence and weakly $G_{h}-$ G h − almost convergence are identical. Finally, some applications of the Orlicz–Pettis theorem with the newly formed sequence spaces and unconditionally Cauchy series $\sum_{k}z_{k}$ ∑ k z k in Y are given.

General Solution and Stability of Additive-Quadratic Functional Equation in IRN-Space

The investigation of the stabilities of various types of equations is an interesting and evolving research area in the field of mathematical analysis. Recently, there are many research papers published on this topic, especially additive, quadratic, cubic, and mixed type functional equations. We propose a new functional equation in this study which is quite different from the functional equations already dealt in the literature. The main feature of the equation dealt in this study is that it has three different solutions, namely, additive, quadratic, and mixed type functions. We also prove that the stability results hold good for this equation in intuitionistic random normed space (briefly, IRN-space).

Some Properties of Fuzzy Inner Product Space

Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement of D.