Inner characterizations of weakly compactly generated Banach spaces and their relatives

We give characterizations of weakly compactly generated spaces, their subspaces, Vašák spaces, weakly Lindelöf determined spaces as well as several other classes of Banach spaces related to uniform Gâteaux smoothness, in terms of the presence of a total subset of the space with some additional properties. In addition, we describe geometrically, when possible, these classes by means of suitable smoothness or rotundity of the norm. As a consequence, we get new, functional analytic proofs of several theorems on (uniform) Eberlein, Gul'ko and Talagrand compacta.     

The McShane integral in weakly compactly generated spaces

Di Piazza and Preiss asked whether every Pettis integrable function defined on [0,1] and taking values in a weakly compactly generated Banach space is McShane integrable. In this paper we answer this question in the negative. Moreover, we give a counterexample where the target Banach space is reflexive.     

Some remarks on weakly compactly generated banach spaces

A simple example is given of a non WCG space whose dual is a WCG space with an unconditional basis. It is proved that ifX* is WCG andX is a subspace of a WCG space thenX itself is WCG. The first named author was supported in part by NSF GP-33578. An erratum to this article is available at .     

Weakly compactly generated Banach spaces and the strong law of large numbers

IfB is a weakly compactly generated Banach space andf: (S,S, ) satisfies the strong law of large numbers, thenf=f ₁+f ₂, wheref ₁ is Bochner -integrable andf ₂ is Pettis -integrable with Pettis norm 0. The decomposition is unique.     

Smoothness and estimates of sequences in Banach spaces

An upper bound for the order of smoothness of bump functions in Banach spaces without copy ofc ₀ is found in terms of lower and upper estimates of their sequences. It is also shown that everyC ^∞-smooth Banach space with symmetric basis either containsc ₀ or is isomorphic toℓ _2n for some integern. Partially supported by DGICYT grant PB 90-0044.     

Integration in Hilbert generated Banach spaces

The purpose of this note is to study the bounded isometry conjecture proposed by Lalonde and Polterovich. In particular, we show that the conjecture holds for the Kodaira-Thurston manifold with the standard symplectic form and for the 4-torus with all linear symplectic forms.     

On weakly null FDD'S in Banach spaces

In this paper we show that every sequence (F _n ) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be “refined” to yield an F.D.D. (G _n ), still having increasing dimensions, so that either every bounded sequence (x _n ), withx _n ∈G _n forn∈ℕ, is weakly null, or every normalized sequence (x _n ),withx _n ∈G _n forn∈ℕ, is equivalent to the unit vector basis of ℓ₁. Crucial to the proof are two stabilization results concerning Lipschitz functions on finite dimensional normed spaces. These results also lead to other applications. We show, for example, that every infinite dimensional Banach spaceX contains an F.D.D. (F _n ),with lim _n→∞dim(F _n )=∞, so that all normalized sequences (x _n ),withx _n ∈F _n ,n∈∕, have the same spreading model overX. This spreading model must necessarily be 1-unconditional overX. Research partially supported by NSF DMS-8903197, DMS-9208482, and TARP 235. Research partially supported by NSF.     

Determinacy and weakly Ramsey sets in Banach spaces

We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper ``Weakly Ramsey sets in Banach spaces.'

     

Fixed points for weakly inward mappings in Banach spaces

S. Hu and Y. Sun [S. Hu, Y. Sun, Fixed point index for weakly inward mappings, J. Math. Anal. Appl. 172 (1993) 266-273] defined the fixed point index for weakly inward mappings, investigated its properties and studied the fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we continue to investigate boundary conditions, under which the fixed point index for the completely continuous and weakly inward mapping, denoted by i(A,Ω,P), is equal to 1 or 0. Correspondingly, we can obtain some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations Ax=μx, which extend many famous theorems such as Leray-Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings. In addition, our conclusions and methods are different from the ones in many recent works.     

LIL behavior for weakly dependent random variables in Banach spaces

Given a sequence of identically distributed ψ-mixing random variables {X _n ; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

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T contains all weakly Lindelöf Banach spaces;

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l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

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T is stable under weak homeomorphisms;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Smoothness of the distribution of the norm in uniformly convex Banach spaces

Let (E, · ) be a uniformly convex Banach space of power type. In this paper we investigate differentiability properties of the distribution function of the norm of a random series with one-dimensional independent components inE.     

Complementation and decompositions in some weakly Lindelöf Banach spaces

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c₀(Γ) or (2) for every c₀(Γ)⊆X has a complemented c₀(E) for an uncountable E⊆Γ or (3) has a decomposition X=A⊕B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P-ideal dichotomy, for every c₀(Γ)⊆C(K) there is a complemented c₀(E) for an uncountable E⊆Γ, which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C(K) for K of height ω+1 where every operator is of the form cI+S for c∈R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K).     

Some estimates for the weakly convergent sequence coefficient in Banach spaces

We present relations between the weakly convergent sequence coefficient of a Banach space and other coefficients. Some estimates are sharp.     

On compactly dominated spaces

Acta Mathematica Hungarica -