Banach spaces with many boundedly complete basic sequences failing PCP

We prove that there exist Banach spaces not containing ?₁, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the problem of Remark 2 in Rosenthal (2007) [12].     

Unconditional martingale difference sequences in Banach spaces

For a Banach space Y, the question of whether Lp(μ,Y) has an unconditional basis if 1<p<∞ and Y has unconditional basis, stood unsolved for a long time and was answered in the negative by Aldous. In this work we prove a weaker, positive result related to this question. We show that if (yj) is a basis of Y and (di) is a martingale difference sequence spanning Lp(μ) then the sequence (di⊗yj) is a basis of Lp(μ,Y) for 1?p<∞. Moreover, if 1<p<∞ and (yj) is unconditional then (di⊗yj) is strictly dominated by an unconditional tensor product basis. In addition, for 1<p<∞, we show that if (di)⊂Lp(μ) is a martingale difference sequence then there exists a constant K>0 so that

     

Boundedly complete basic sequences,c 0-subspaces,and injections of Banach spaces

We study the connection between topological properties of subsets of a given Banach space and their images under linear, continuous one-to-one mappings on the one hand and the existence in a given Banach space of either a boundedly complete basic sequence (BCBS) or an isomorphic copy ofc _o (c _o -subspace) on the other hand. We present criteria for the existence of a BCBS. They are deduced from new characterisations ofG _δ-embeddings which we also present. We obtain a necessary and sufficient condition for separability of a dual Banach space in terms of saturation by BCBS. Criteria for the existence in a Banach space of ac _o -subspace are also presented. We describe the class of separable Banach spaces which contains either a BCBS or ac _o -subspace. This research was supported by the Rashi Foundation.     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

For a blockwise martingale difference sequence of random elements {Vn , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form limn→∞∑ n i=1 Vi /gn = 0 almost surely to hold where the constants gn ↑∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.     

Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.     

Some Characterizations of Hereditarily Indecomposable Banach Spaces

In this paper we first discuss the relations between some G-M-type spaces, and the previous eight kinds of G-M-type Banach spaces are merged into four different kinds. Then we build a Generalized Operator Extension Theorem, and introduce the concept of complete minimal sequences. Some sufficient and necessary conditions under which a Banach space is a hereditarily indecomposable space are given. Finally, we give some characterizations of hereditarily indecomposable Banach Spaces.     

Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

Diagonal sequence property in Banach spaces with weaker topologies

We investigate the diagonal sequence property in Banach spaces with weaker topologies. In particular, we present examples of Banach spaces with weaker locally convex topologies which have the diagonal sequence property but are not Fréchet–Urysohn. The examples answer negatively a question of Averbukh and Smolyanov. We give also a very simple proof of the fact that each Banach space contains a subset A whose weak closure includes 0, but 0 is not contained in the weak closure of any bounded subset of A.     

Multi-quadratic mappings in Banach spaces

We prove the stability of multi-quadratic functional equations in Banach spaces.

     

Separated sequences in nonreflexive Banach spaces

We prove that there is 1$"> such that the unit ball of any nonreflexive Banach space contains a -separated sequence. The supremum of these constants is estimated from below by and from above approximately by . Given any 1$">, we also construct a nonreflexive space so that if the convex hull of a sequence is sufficiently close to the unit sphere, then its separation constant does not exceed .

     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Almost overcomplete and almost overtotal sequences in Banach spaces

We introduce the new concepts of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we present several applications of this fact.     

Discrete chaos in Banach spaces

This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiate map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.     

B值随机元阵列的矩完全收敛性

在随机元阵列随机有界于某非负随机变量的条件下,得到了B值行独立的随机元阵列的矩完全收敛性的一些充分条件.同时研究了p型Banach空间中行独立的随机元阵列的矩完全收敛性.     

On the Davis inequality in Banach function spaces

We give a characterization of those Banach function spaces in which the Davis inequality for martingales is valid. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Urysohn space embeds in Banach spaces in just one way

In a Master's thesis in 1985 and a subsequent paper published in 1992, the author discovered that the universal separable metric space (up to isometry) discovered by Urysohn in 1925 has a uniquely determined linear closure (up to linear isometry) when isometrically embedded in a Banach space so as to include the zero of the Banach space. The proof of this result is given in this note and the current status of some related questions is discussed.     

Open sets with the Runge property in Banach spaces

We show that every pseudoconvex open subset of a Banach space with an unconditional Schauder basis has the F-valued Runge property for every Banach space F.