Some Directions of Research in Banach Space Theory

Abstract

This chapter discusses the different aspects of the researches in Banach space theory. One of the main aims of Banach space theory is their classification. Spaces can be classified according to different points of view and this choice affects considerably the outcome as well as the methods of investigation. The most elementary way of classifying Banach spaces is to consider only their topological structure. One would like to determine those pairs of Banach spaces that, as topological spaces with respect to their norm topologies, are homeomorphic to each other. The solution to this problem is completely obvious in the finite dimensional case and two such spaces are homeomorphic if and only if they have the same. This condition is necessary also in the infinite dimensional case in the sense that if two general Banach spaces are homeomorphic to each other then they have the same density character. The density character of a Banach space is its unique topological invariant. It is found that a positive solution would be obtained if one could show that every Banach space contains a subspace with an unconditional basis of infinite length.