A local metric characterization of Banach spaces
A modulus for property (β) of Rolewicz
We define a modulus for the property (β) of Rolewicz and study some useful properties in fixed point theory for nonexpansive mappings. Moreover, we calculate this modulus in spaces for the main measures of noncompactness.
A Natural Class of Sequential Banach Spaces
We introduce and study a natural class of variable exponent spaces, which generalizes the classical spaces and c₀. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.
A nearly uniformly convex space which is not a -space
A new approach to several results of V. Milman.
A new characterization of Eberlein compacta
We give a sufficient and necessary condition for a Radon-Nikodým compact space to be Eberlein compact in terms of a separable fibre connecting weak-* and norm approximation.
A new class of weakly countably determined Banach spaces
A class of Banach spaces, countably determined in their weak topology (hence, WCD spaces) is defined and studied; we call them strongly weakly countably determined (SWCD) Banach spaces. The main results are the following: (i) A separable Banach space not containing ℓ¹(ℕ) is SWCD if and only if it has separable dual; thus in particular, not every separable Banach space is SWCD. (ii) If K is a compact space, then the space C(K) is SWCD if and only if K is countable.
A new convexity property that implies a fixed point property for
In this paper we prove a new convexity property for L₁ that resembles uniform convexity. We then develop a general theory that leads from the convexity property through normal structure to a fixed point property, via a theorem of Kirk. Applying this theory to L₁, we get the following type of normal structure: any convex subset of L₁ of positive diameter that is compact for the topology of convergence locally in measure, must have a radius that is smaller than its diameter. Indeed, a stronger result...
A New Dichotomy for Banach Spaces.
A new proof of James' sup theorem.
We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson: "If a normed space E does not contain any asymptotically isometric copy of l1, then every bounded sequence of E' has a normalized l1-block sequence pointwise converging to 0".
A new result of G. A. Edgar on representing points in a convex bounded subset of Banach spaces with the Radon-Nikodym property as barycentres of Radon measures
A new type of orthogonality for normed planes
In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions .
A nondentable set without the tree property
A normalized weakly null sequence with no shrinking subsequence in a Banach space not containing
A note about spaces.
A note on close-to-normal structure
A note on continuous restrictions of linear maps between Banach spaces.
A Note on cotype of smooth spaces.
A note on Dunford-Pettis like properties and complemented spaces of operators
Equivalent formulations of the Dunford-Pettis property of order (), , are studied. Let , , , , and denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and -convergent operators from to . Classical results of Kalton are used to study the complementability of the spaces and in the space , and of in and .
A note on lattice renormings
It is shown that every strongly lattice norm on can be approximated by smooth norms. We also show that there is no lattice and Gâteaux differentiable norm on .