Around finite-dimensionality in functional analysis: a personal perspective

As objects of study in functional analysis, Hilbert spaces stand out as special objects of study as do nuclear spaces (a'la Grothen-dieck) in view of a rich geometrical structure they possess as Banach and Frechet spaces, respectively. On the other hand, there is the class of Banach spaces including certain function spaces and sequence spaces which are distinguished by a poor geometrical structure and are subsumed under the class of so-called Hilbert-Schmidt spaces. It turns out that these three classes of spaces are mutually disjoint in the sense that they intersect precisely in finite dimensional spaces. However, it is remarkable that despite this mutually exclusive character, there is an underlying commonality of approach to these disparate classes of objects in that they crop up in certain situations involving a single phenomenon—the phenomenon of finite dimensionality—which, by definition, is a generic term for those properties of Banach spaces which hold good in finite dimensional spaces but fail in infinite dimension.