Separable Banach spaces which admitl n ∞ approximations

In this paper we study a class of separable Banach spaces which can be approximated by certain special finite-dimensional subspaces. This class is characterized in Theorem 1.1, from which it follows that the space of continuous scalar-valued functions on a compact metric space always belongs to this class, and that every member of this class has a monotone basis. Supported in part by N.S.F. Grant 11-5020. Supported in part by N.S.F. Grant GP-3579.     

On shrinking and boundedly complete Schauder frames of Banach spaces

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.     

Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

Construction of best Bregman approximations in reflexive Banach spaces

An iterative method is proposed to construct the Bregman projection of a point onto a countable intersection of closed convex sets in a reflexive Banach space.

     

Banach spaces of distributions having two module structures

After a discussion of a space of test functions and the corresponding space of distributions, a family of Banach spaces (B, ∥ ∥_B) in standard situation is described. These are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution. The main results concern relations between the different spaces associated to B established by means of well-known methods from the theory of Banach modules, among them B₀ and $\text{B}\text{?}$, the closure of the test functions in B and the weak relative completion of B, respectively. The latter is shown to be always a dual Banach space. The main diagram, given in Theorem 4.7, gives full information concerning inclusions between these spaces, showing also a complete symmetry. A great number of corresponding formulas is established. How they can be applied is indicated by selected examples, in particular by certain Segal algebras and the A_p-algebras of Herz. Various further applications are to be given elsewhere.     

Relations between rational and polynomial approximations in Banach spaces

В РАБОтЕ ИжУЧАУтсь Ус лОВИь, ДАУЩИЕ РАВЕНст ВА ИлИ стРОгИЕ НЕРАВЕНстВА МЕжДУ ВЕлИЧИНАМИ НАИ МЕНьшИх РАцИОНАльНы х И пОлИНОМИАльНых УклО НЕНИИ В БАНАхОВых пРОстРАНс тВАх. лль ОпРЕДЕлЕННО гО клАссА БАНАхОВых пРОстРАНс тВ НАИДЕНы ОгРАНИЧЕНИь НА ВОжМОжНОЕ кОлИЧЕс тВО НЕтРИВИАльНых сОВпА ДЕНИИ пОлИНОМИАльНы х И РАцИОНАльНых АппРО ксИМАцИИ РАВНых стЕп ЕНЕИ. РАссМОтРЕНы пРИБлИж ЕНИь пОДпРОстРАНстВАМИ Р АцИОНАльНых ФУНкцИИ с ФИксИРОВАННыМ “жНАМ ЕНАтЕлЕМ”. пОлУЧЕНы НЕОБхОДИМы Е УслОВИь Дль РАцИОНА льНОИ ФУНкцИИ НАИлУЧшЕгО пРИБлИжЕНИь. В жАклУЧ ИтЕльНОИ ЧАстИ ДОкАж АНА ВОжМОжНОсть сОВпАДЕ НИь НЕ РАВНых НУльУ ВЕлИЧИН НАИлУЧшИх пОлИНОМИА льНых стЕпЕНИn И РАцИОНАль Ных стЕпЕНИ (n, 1) пРИБлИжЕНИ И В пРОстРАНстВЕ хАРД ИH ₂(D).     

Musielak–Orlicz BMO-type Spaces Associated with Generalized Approximations to the Identity

Let X be a space of homogenous type and ψ: X × [0,∞) → [0,∞) be a growth function such that ψ(·,t) is a Muckenhoupt weight uniformly in t and ψ(x,·) an Orlicz function of uniformly upper type 1 and lower type p ∈(0,1].In this article,the authors introduce a new Musielak–Orlicz BMO-type space BMOψA(X) associated with the generalized approximation to the identity,give out its basic properties and establish its two equivalent characterizations,respectively,in terms of the spaces BMOψA,max(X) and BMOψA(X).Moreover,two variants of the John–Nirenberg inequality on BMOψA(X) are obtained.As an application,the authors further prove that the space BMOψΔ1/2(Rn),associated with the Poisson semigroup of the Laplace operator Δ on Rn,coincides with the space BMOψ(Rn) introduced by Ky.     

Banach spaces having the Radon-Nikodym property and numerical index 1

Let be a Banach space with the Radon-Nikodym property. Then, the following are equivalent.

(i) has numerical index 1.

(ii) for all and . (iii) is an almost-CL-space.

(iv) There are a compact Hausdorff space and a linear isometry such that for all and .

If is a real space, the above conditions are equivalent to being semi-nicely embedded in some space .

     

The split common null point problem and the shrinking projection method in Banach spaces

In this paper, we consider the split common null point problem with resolvents of maximal monotone operators in Banach spaces. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces.     

Linear approximations for differential equations in Banach spaces and Lyapunov stability

Doklady Mathematics -     

Weakly continuous holomorphic functions on Banach spaces with a shrinking and unconditional basis

Let E and F be complex Banach spaces, and let U be an open ball in E. We show that if E has a shrinking and unconditional basis, then every holomorphic function that is weakly continuous on U-bounded sets is weakly uniformly continuous on U-bounded sets.     

A shrinking projection method for solving the split common null point problem in Banach spaces

Numerical Algorithms - In this paper, in order to solve the split common null point problem, we investigate a new explicit iteration method, base on the shrinking projection method and...     

Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces

In this paper, we introduce and investigate a new class of mixed quasi-variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to Ding–Tan and a lemma due to Chang, we establish some existence and uniqueness results of solution for the mixed quasi-variational-like inequality. Next, by using a KKM theorem due to Fan and an auxiliary principle technique due to Cohen, we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms. Our results extend, improve and unify several known results in the literature.     

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω₁ of weight ℵ₁, such that the space C(Kω₁) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω₁ is a continuous image of a Valdivia compact and every separable subspace of C(Kω₁) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.     

Viscosity approximations by generalized contractions for resolvents of accretive operators in Banach spaces

In this paper, we prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method with a generalized contraction mapping. The proximal point algorithm in a Banach space is also considered. The results extend some very recent theorems of W. Takahashi.     

A consturction of monotonically convergent sequences from successive approximations in certain Banach spaces

Summary LetA: XX (whereX=C ^q [a, b] orL ^p [a, b]) be a contraction having the fixed pointf. In this note, using ideas from [1–8], we obtain a modified successive approximation sequence which approximatesf and which has certain properties regarding monotonicity too.     

Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces

The existence of common fixed points is established for three mappings where T is either generalized (f,g)-nonexpansive or asymptotically (f,g)-nonexpansive on a nonempty subset of a Banach space. As applications, the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained. Our results unify and substantially improve several recent results existing in the current literature.