Kadec–Klee property and fixed points

Let X be a reflexive Banach space which does not have the Kadec–Klee property. Then there exists a compact mapping f   from the unit ball _BX$B_X$ of X   to the dual space X^?$X^{?}$ such that _infx∈BX‖f(x)‖>0${\inf }_{x\in B_X}\Vert f(x)\Vert >0$ and 〈f(x),x〉<‖f(x)‖$〈f(x),x〉<\Vert f(x)\Vert$ for every x∈_BX$x\in B_X$.     

Non-expansive Mappings and Iterative Methods in Uniformly Convex Banach Spaces

In this article, we will investigate the properties of iterative sequence for non-expansive mappings and present several strong and weak convergence results of successive approximations to fixed points of non-expansive mappings in uniformly convex Banach spaces. The results presented in this article generalize and improve various ones concerned with constructive techniques for the fixed points of non-expansive mappings.     

Kadec–Klee properties of some quasi-Banach function spaces

We study Kadec–Klee properties with respect to global (local) convergence in measure in quasi-Banach function spaces. First, we prove some general results which can be of independent interest. Next, we investigate these properties in symmetrizations \(E^{\left( *\right) }\). Finally, we apply general results to study these properties in Marcinkiewicz and Lorentz spaces.     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

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.     

L~P-orthogonality in Banach Spaces

The notion of L~P-orthogonality was posed by F. Sullivan [1], but it hadnever been studied. The purpose of this paper is to give some results concerningthe L~P-orthogonality in Banach spaces.     

Banach—Saks Property and Property (β) in Cesàro Sequence Spaces

A new constant C(X) for any Banach space X is introduced. It is proved that C(X) < 2 implies the weak Banach—Saks property for the space X. In particular, C(ces_p) is found for Cesàro sequence space ces_p (1 < p < ). Moreover, it is shown that the space ces_p (1 < p < ) has property ().AMS Subject Classification (2000), 46E30, 46E40, 46B20Supported by NSF and ECF of China.     

Stampfli''''s Theorem for Operators on Some Banach Spaces

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Differentiability of Lipschitz Maps from Metric Measure Spaces to Banach Spaces with the Radon–Nikodym Property

We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.     

Von Neumann–Schatten Frames in Separable Banach Spaces

In this paper we introduce the notion of a von Neumann-Schatten p-frame in separable Banach spaces and obtain some of their characterizations. We show that p-frames and g-frames are a class of von Neumann-Schatten p-frames.     

Exposed and Strongly Exposed Points of Finite Order in Some Banach Spaces

in this paper we study the exposed and strongly exposed points of finite order inl(D,X) as well as in L(ll (D,X), Y) and L(X, l(D,Y) ), where X and Y are real Banachspaces while D is a non empty index set. In particular, we characterize the exposed andstrongly exposed points in these spaces.     

The Coarse ℓ p -Novikov Conjecture and Banach Spaces with Property (H)∗

In this paper, for 1 < p < ∞, the authors show that the coarse ? p -Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu’s Property (H).     

On the WAε Property and g-NUCε Banach Spaces

The WAε property and g-NUCε, g-NUC Banach spaces are introduced. We prove that the WAε property is equivalent to the WBS property and the g-NUCε (resp., g-NUC) spaces are equivalent to the NUCε (resp. , NUC) spaces possessing the BS property. So we obtain a characterization of the NUCε (resp, , NUC) spaces possessing the BS property.     

The Radon–Nikodym Property for Tensor Products of Banach Lattices

Let 1 ≤ p < ∞. We show that , the Fremlin projective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that , the Wittstock injective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from ℓ_p' to X is compact, where 1/p +1/p'= 1 and let ℓ_p' = c₀ if p = 1. The author gratefully acknowledges support from the Office of Naval Research Grant # N00014-03-1-0621     

The Marcinkiewicz–Zygmund LLN in Banach Spaces: A Generalized Martingale Approach

A result due to Gut asserts that the Marcinkiewicz–Zygmund strong law of large numbers for real-valued random variables is an amart a.s. convergence property. In this paper, a necessary and sufficient condition is given, under which that SLLN is also a quasimartingale. We also study the case of Banach-space valued r.v. and show that the scalar result remains true when the space is of suitable stable type.     

Implicit Complementarity Problems in Banach Spaces

We utilize the method of multivalued fixed point in this paper to prove the existence theorems of solutions for some implicit complementarity problems in Banach spaces.     

A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X. (Received 3 March 2000; in revised form 29 June 2000)     

A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X.     

A New System of Generalized Variational Inclusions Involving $H$-$η$-Monotone Operators in Uniformly Smooth Banach Spaces

In this paper, we introduce and study a new system of generalized variational inclusions involving $H$-$η$-monotone operators in uniformly smooth Banach spaces. Using the resolvent operator technique associated with $H$-$η$-monotone operators, we prove the approximation solvability of solutions using an iterative algorithm. The results in this paper extend and improve some known results from the literature.