A Weak Groethendieck Compactness Principle for Infinite Dimensional Banach Spaces

We introduce and investigate the notion of $p$-convergence in a Banach space. Among others, a Grothendieck-like result is obtained; namely, a subset of a Banach space is relatively $p$-compact if and only if it is contained in the closed convex hull of a $p$-null sequence. We give a description of the topological dual of the space of all $p$-null sequences which is used to characterize the Banach spaces enjoying the property that every relatively $p$-compact subset is relatively $q$-compact ($1\leq q<p$). As an application, Banach spaces satisfying that every relatively $p$-compact set lies inside the range of a vector measure of bounded variation are characterized.

On super fixed point property and super weak compactness of convex subsets in Banach spaces

Sequence spaces are one of the topics in the field functional analysis such as null sequence space, converging sequence space and bounded sequence space designated and respectively. Those sequence spaces are normed spaces. Entry of Copson matrices denoted by with value and 0 for other . Null sequence space, , converging sequence space and bounded sequence space with Copson matrices built new sequence spaces that are squence spaces . The purpose of this study is to get some properties that occur to those sequence spaces with the same normed as sequence space . The results of this study are sequence spaces are Banach spaces with mapping coordinate is continues (BK), there are inclusion properties on sequence spaces, sequence space has AK property and sequence spaces and are compact.

For the duality mapping on a Banach space the relation between lower semi-continuity and upper semi-continuity properties is explored, upper semi-continuity is characterized in terms of slices of the ball and upper semi-continuity properties are related to geometrical properties which imply that the space is an Asplund space. The duality mapping is a natural set-valued mapping from the unit sphere of a normed linear space into subsets of its dual sphere, and which for an inner product space is the mapping associating an element of the unit sphere with the corresponding continuous linear functional given by the inner product. It is an example of a subdifferential mapping of a continuous convex function (in this case, the norm), which is in turn a special kind of maximal monotone mapping. Cudia [4, p. 298] showed that the duality mapping is always upper semi-continuous when the space has the norm and the dual space has the weak* topology, and Kenderov [10, p. 67] extended this to maximal monotone mappings. Bonsall, Cain, and Schneider [3] used the property to prove the connectedness of the numerical range of an operator on a normed linear space. Along with the activity which culminated in StegalΓs theorem [15] characterizing an Asplund space as one whose dual has the Radon-Nikodym property, there has been some interest in finding geometrical conditions sufficient for a space to be Asplund. A Banach space X is an Asplund space if every continuous convex function defined on an open convex subset of X is strongly differentiate on a dense Gδ subset of its domain. Ekeland and Lebourg [6, p. 204] have shown that a Banach space is Asplund if there exists a strongly differentiate real function on the space with bounded nonempty support, in particular, if the space can be equivalently renormed to have norm strongly differentiable on the unit sphere. Using StegalΓs theorem, a result of Diestel and Faires [5, p. 625] gives that a Banach space is Asplund if the space can be equivalently renormed to be very smooth, that is, to be smooth and to have the single valued duality mapping continuous when the space has the norm and the dual space has the weak topology. Recently Smith and Sullivan [13, Theorem 15] have exhibited a more general condition which is sufficient for a Banach space to be Asplund. We show that

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.

Chapter 52 - The interplay between compact spaces and the Banach spaces of their continuous functions

SynopsisThe compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence, which combines with any discretization on the spatial domain. We show optimality properties of this solver, and present numerical computations. Introduction Stochastic Galerkin methods have emerged in the past decade as an efficient solution procedure for boundary value problems depending on random data, see [DBO01, XK02, BTZ04, WK05, MK05, FST05, WK06, TS07, BS09, BAS10]. These methods approximate the random solution by a Galerkin projection onto a finite dimensional space of random fields. This requires the solution of a single coupled system of deterministic equations for the coefficients of the Galerkin projection with respect to a predefined set of basis functions on the parameter domain. A major remaining obstacle is the construction of suitable spaces in which to compute approximate solutions. These should be adapted to the stochastic structure of the equation. Simple tensor product constructions are infeasible due to the high dimensionality of the parameter domain in case of input random fields with low regularity. Parallel to but independently from the development of stochastic Galerkin methods, a new class of adaptive methods has emerged, which are set not in the continuous framework of a boundary value problem, but rather on the level of coefficients with respect to a hierarchic Riesz basis, such as a wavelet basis. Due to the norm equivalences constitutive of Riesz bases, errors and residuals in appropriate sequence spaces are equivalent to those in physically meaningful function spaces. This permits adaptive wavelet methods to be applied directly to a large class of equations, provided that a suitable Riesz basis is available. For symmetric elliptic problems, the error of the Galerkin projection onto the span of a set of coefficients can be estimated using a sufficiently accurate approximation of the ∗Research supported in part by the Swiss National Science Foundation grant No. 200021-120290/1.

Weakly compact composition operators on vector-valued BMOA

Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L1 (μ,X) the Banach space of (equivalence classes of) μ-strongly measurable X-valued Bochner integrable functions f:Ω→X normed byThe problem of characterizing the relatively weakly compact subsets of L1(Ω, X) remains open. It is known that for a bounded subset of L1(μ, X) to be relatively weakly compact it is necessary that the set be uniformly integrable; recall that K ⊆ L1, (μ, X) is uniformly integrable whenever given ε &gt;0 there exists δ &gt; 0 such that if μ (E) ≦ δ then ∫E∥f∥ dμ ≦ δ, for all f ∈ K. S. Chatterji has noted that in case X is reflexive this condition is also sufficient [4]. At present unless one assumes that both X and X* have the Radon-Nikodym Property (see [1]), a rather severe restriction which, for purposes of potential applicability, is tantamount to assuming reflexivity, no good sufficient conditions for weak compactness in L1(μ, X) exist. This note puts forth such sufficient conditions; the basic tool is the recent factorization method of W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczynski [3].

k-2 - Banach Spaces and Topology (II)

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

In questo lavoro presentiamo alcuni risultati relativi alla debole compattezza e alla compattezza degli operatori dominati sugli spaziC(Ω,X).

Hagler and the first named author introduced a class of hereditarily l 1 Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l p Banach spaces for 1 ⩽ p < ∞. Here we use these spaces to introduce a new class of hereditarily l p (c 0) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l 1 Banach spaces failing the Schur property.

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.