Theoretical Statistics and Mathematics Unit, ISI Delhi
A property of Banach spaces shall be designated as a finite dimensional property (FD-property) if it holds good for all finite dimensional Banach spaces but fails in each infinite dimensional Banach space. However, in certain situations it becomes necessary to ‘salvage’ these (FD)-properties in an infinite dimensional setting that include, in particular, the class of Frechet spaces. As one of (three) important features of (FD-properties, it is possible to propose a suitable analogue of a given (FD)-property (P) in an infinite dimensional context and then completely describe the class of Frechet spaces witnessing the validity of property (P). What is indeed remarkable is that in a large number of cases, the class of Frechet spaces that result in the process are precisely those which are nuclear in the sense of Grothendieck.
For the purpose of this talk, this will be shown indeed to be the case in the context of certain (FD)-properties arising within the theory of vector measures. It is also intended, within the constraints of time, to briefly talk about some open problems surrounding this circle of ideas and involving, in particular, a certain operator-analogue of the Hahn-Banach theorem on the one hand and convergence of ‘signed’ series on the other.