Let U, V be two symmetric convex bodies in and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors such that, for each choice of signs , one has where . Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence such that the series is divergent for any choice of signs and any permutation π of indices.
What follows is the opening conference of the late night seminar at the III Conference on Banach Spaces held at Jarandilla de la Vera, Cáceres. Maybe the reader should not take everything what follows too seriously: after all, it was designed for a friendly seminar, late in the night, talking about things around a table shared by whisky, preprints and almonds. Maybe the reader should not completely discard it. Be as it may, it seems to me by now that everything arrives in the nick of time. A twisted...
In this article we formalize one of the most important theorems of linear operator theory - the Closed Graph Theorem commonly used in a standard text book such as [10] in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.
It is proved that for any Banach space X property (β) defined by Rolewicz in [22] implies that both X and X* have the Banach-Saks property. Moreover, in Musielak-Orlicz sequence spaces, criteria for the Banach-Saks property, the near uniform convexity, the uniform Kadec-Klee property and property (H) are given.
The talk presented a survey of results most of which have been obtained over the last several years in collaboration with M.Florencio and P.J.Paúl (Seville). The results concern the question of barrelledness of locally convex spaces equipped with suitable Boolean algebras or rings of projections. They are particularly applicable to various spaces of measurable vector valued functions. Some of the results are provided with proofs that are much simpler than the original ones.