A bornological approach to rotundity and smoothness applied to approximation.
A characterization of complex -preduals via a complex barycentric mapping
We provide a complex version of a theorem due to Bednar and Lacey characterizing real -preduals. Hence we prove a characterization of complex -preduals via a complex barycentric mapping.
A C(K) Banach space which does not have the Schroeder-Bernstein property
We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...
A class of Banach sequence spaces analogous to the space of Popov
Hagler and the first named author introduced a class of hereditarily Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily Banach spaces for . Here we use these spaces to introduce a new class of hereditarily Banach spaces analogous of the space of Popov. In particular, for the spaces are further examples of hereditarily Banach spaces failing the Schur property.
A conjecture about the Dunford-Pettis property
A continuum of totally incomparable hereditarily indecomposable Banach spaces
A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.
A counterexample to a complementation problem
A counterexample to several questions about Banach spaces
A function space C(K) not weakly homeomorphic to C(K)×C(K)
A geometrical/combinatorical question with implications for the John-Nirenberg inequality for BMO functions
The first and last sections of this paper are intended for a general mathematical audience. In addition to some very brief remarks of a somewhat historical nature, we pose a rather simply formulated question in the realm of (discrete) geometry. This question has arisen in connection with a recently developed approach for studying various versions of the function space BMO. We describe that approach and the results that it gives. Special cases of one of our results give alternative proofs of the...
A nondentable set without the tree property
A normalized weakly null sequence with no shrinking subsequence in a Banach space not containing
A Note on compactness in Banach 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 extreme contractions on -spaces
A note on isomorphisms between powers of Banach spaces.
A note on natural tensor products containing complemented copies of
A note on norm-attaining functionals
A note on the paper of Shütt "Unconditionality in tensor products''
A quasi-dichotomy for C(α,X) spaces, α < ω₁
We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm. Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true. (1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X. (2) For any infinite countable...