Multi-normed Spaces Research Articles

Free and projective generalized multinormed spaces

The paper investigates free and projective L-spaces, where L is a given normed space. These spaces form a far-reaching generalization of known p-multinormed spaces; in particular, if L=Lp(X), the L-spaces can be considered as p-multinormed spaces, based on arbitrary σ-finite measure spaces X (for “canonical” p-multinormed spaces, X=N with the counting measure). We first describe a “naturally appearing” functor, based on paving L with contractively complemented finite dimensional subspaces. This finite dimensionality is essential; it permits us to describe a free L-space for this functor. As a corollary, we obtain a wide variety of projective L-spaces. For “nice” spaces L (such as the space of simple p-integrable functions on a measure space), we obtain a full characterization of projective L-spaces; as a particular case, we recover a description of projective p-multinormed spaces.

Local stability of mappings on multi-normed spaces

First we investigate the Hyers–Ulam stability of the Cauchy functional equation for mappings from bounded (unbounded) intervals into Banach spaces. Then we study the Hyers–Ulam stability of the functional equation f(xy)=xg(y)+h(x)y for mappings from bounded (unbounded) intervals into multi-normed spaces.

Multi-normed spaces based on non-discrete measures and their tensor products

Lambert discovered a new type of structures situated, in a sense, between normed spaces and abstract operator spaces. His definition was based on the notion of amplifying a normed space by means of the spaces . Later, several mathematicians studied more general structures (`-multi- normed spaces') introduced by means of the spaces , . We pass from to with an arbitrary measure. This becomes possible in the framework of the non- coordinate approach to the notion of amplification. In the case of a discrete counting measure, this approach is equivalent to the approach in the papers mentioned. Two categories arise. One consists of amplifications by means of an arbitrary normed space, and the other consists of -convex amplifications by means of . Each of them has its own tensor product of objects (the existence of each product is proved by a separate explicit construction). As a final result, we show that the `-convex' tensor product has an especially transparent form for the minimal -amplifications of -spaces, where is conjugate to . Namely, tensoring and , we obtain .

On the stability of the Pexiderized cubic functional equation in multi-normed spaces

In this paper, we investigate the Hyers-Ulam stability of the orthogonally cubic equation and Pexiderized cubic equation [f(kx+y)+f(kx-y)=g(x+y)+g(x-y)+frac{2}{k}g(kx)-2g(x),]in multi-normed spaces by the direct method and the fixed point method. Moreover, we prove the Hyers-Ulam stability of the $2$-variables cubic equation [ f(2x+y,2z+t)+f(2x-y,2z-t) =2f(x+y,z+t) +2f(x-y,z-t)+12f(x,z),]and orthogonally cubic type and $k$-cubic equation in multi-normed spaces. A counter example for non stability of the cubic equation is also discussed.

Stability and nonstability of octadecic functional equation in multi-normed spaces

In this paper, we introduce octadecic functional equation. Moreover, we prove the stability of the octadecic functional equation in multi-normed spaces by using the fixed point method.

Approximation of the multiplicatives on random multi-normed space

In this paper, we consider random multi-normed spaces introduced by Dales and Polyakov (Multi-Normed Spaces, 2012). Next, by the fixed point method, we approximate the multiplicatives on these spaces.

On some generalizations of metric, normed, and unitary spaces

Earlier the author offered new notions of multimetric, multinormed and multiunitary spaces which are generalizations of metric, normed and unitary spaces respectively. In this article the properties of these spaces under different topological operations have been considered.

Asymptotic aspect of Jensen and Jensen type functional equations in multi-normed spaces

Asymptotic aspect of Jensen and Jensen type functional equations in multi-normed spaces

A Theory of Summability on a Space of Generalized Functions

A theory of summability of orthonormal sets is introduced in multinormed spaces. The approach which is presented caters for infinite sets , where the index set may be uncountable, and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions. These spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in Hilbert spaces. A space of almost-periodic generalized functions, in which each generalized function can be expanded in terms of an uncountable basis of exponential functions, is obtained as a special case of our theory.

Multi-norms and the injectivity of L p (G )

Let G be a locally compact group, and take p∈(1, ∞). We prove that the Banach left L1(G)-module Lp(G) is injective (if and) only if the group G is amenable. Our proof uses the notion of multi-norms. We also develop the theory of multi-normed spaces.

Multi-normed spaces

We modify the very well known theory of normed spaces (E, ‖ · ‖) within functional analysis by considering a sequence (‖ · ‖n : n ∈ N) of norms, where ‖ · ‖n is defined on the product space En for each n ∈ N. Our theory is analogous to, but distinct from, an existing theory of ‘operator spaces’; it is designed to relate to general spaces Lp for p ∈ [1,∞], and in particular to L1-spaces, rather than to L2-spaces. After recalling some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a ‘multinormed space’ ((En, ‖ · ‖n) : n ∈ N), where (E, ‖ · ‖) is a normed space. Several different, equivalent characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multinorms are the minimum and maximum multi-norm based on a given space. Multi-norms measure ‘geometrically features’ of normed spaces, in particular by considering their ‘rate of growth’. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following standard presentations of the foundations of functional analysis, we consider generalizations to ‘multi-topological linear spaces’ through ‘multi-null sequences’, and to ‘multi-bounded’ linear operators, which are exactly the ‘multi-continuous’ operators. We define a new Banach space M(E,F ) of multi-bounded operators, and show that it generalizes wellknown spaces, especially in the theory of Banach lattices. We conclude with a theory of ‘orthogonal decompositions’ of a normed space with respect to a multi-norm, and apply this to construct a ‘multidual’ space. Applications of this theory will be presented elsewhere. 2000 Mathematics Subject Classification. Primary 43A10, 43A20; secondary 46J10.

ON THE STABILITY OF A JENSEN TYPE FUNCTIONAL EQUATION IN MULTI-NORMED SPACES

In this paper we investigate the Hyers-Ulam stability of a Jensen type functional equation in multi-normed spaces and then extend the result to multi-normed left modules over a normed algebra A.

JENSEN'S FUNCTIONAL EQUATION IN MULTI-NORMED SPACES

We investigate the Hyers-Ulam stability of the Jensen functional equation for mappings from linear spaces into multi-normed spaces. We then establish an asymptotic behavior of the Jensen equation in the framework of multi-normed spaces which are somewhat similar to the operator sequence spaces and have some connections with operator spaces and Banach lattices.

JENSEN'S FUNCTIONAL EQUATION IN MULTI-NORMED SPACES

We investigate the Hyers-Ulam stability of the Jensen functional equation for mappings from linear spaces into multi-normed spaces. We then establish an asymptotic behavior of the Jensen equation in the framework of multi-normed spaces which are somewhat similar to the operator sequence spaces and have some connections with operator spaces and Banach lattices.

Asymptotic aspect of the quadratic functional equation in multi-normed spaces

We investigate the Hyers–Ulam stability of the quadratic functional equation for mappings from abelian groups into multi-normed spaces. We also study the stability on a restricted domain and present an asymptotic behavior of the quadratic equation in the framework of multi-normed spaces.

STABILITY OF MAPPINGS ON MULTI-NORMED SPACES

AbstractIn this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers–Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.