**M.A. Sofi**

Kashmir University, Srinagar

February 18, 2010

**On certain curious aspects of finite-dimensionality phenomena arising in functional analysis**:
A property (P) of Banach spaces is said to be a finite-dimensional
property((FD)-property,for short) if it holds in all finite dimensional Banach spaces but fails in each infinite dimensional Banach space. A classical example of such a property is provided by the compactness of the closed unit ball of a Banach space. At a less trivial level, the equivalence of absolute and unconditional convergence of series in a Banach space provides yet another but perhaps one of the most important examples of this phenomemnon.

We discuss some of the issues arising out of this, in particular, how it necessitates the introduction of Nuclear spaces (due to Grothendieck) on the one hand and to the geometry of Banach spaces on the other.