A single technique provides short proofs of some results about drop properties on locally convex spaces. It is shown that the quasi drop property is equivalent to a drop property for countably closed sets. As a byproduct, we prove that the drop and quasi drop properties are separably determined.
È noto che se uno spazio di Banach è quasi-smooth (cioè, la sua applicazione di dualità è debolmente semicontinua superiormente in senso ristretto), allora il suo duale non ha sottospazi chiusi normanti propri. Inoltre, se uno spazio di Banach ha una norma equivalente la cui applicazione di dualità ha un grafo che contiene superiormente un'applicazione debolmente semicontinua superiormente in senso ristretto, allora lo spazio è Asplund. Dimostriamo che se uno spazio di Banach ha una norma equivalente...
We prove that weakly Lindelöf determined Banach spaces are characterized by the existence of a ``full'' projectional generator. Some other results pertaining to this class of Banach spaces are given.
Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual of a locally convex space is the -closure of the union of countably many -relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.
This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-requisites and get a feeling about this area of Banach space theory. Many open problems of different level of difficulty are discussed. For the reader...
The strong subdifferentiability of norms (i.eȯne-sided differentiability uniform in directions) is studied in connection with some structural properties of Banach spaces. It is shown that every separable Banach space with nonseparable dual admits a norm that is nowhere strongly subdifferentiable except at the origin. On the other hand, every Banach space with a strongly subdifferentiable norm is Asplund.
The dual space of a WUR Banach space is weakly K-analytic.
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