Separable Banach Spaces

In this chapter, we introduce the basic definitions concerning biorthogonal systems in Banach spaces and discuss several results, mostly in the separable setting, related to this structure. When searching for a system of coordinates to represent any vector of a (separable) Banach space, a natural approach is to consider the concept of a Schauder basis. Unfortunately, not every separable Banach space has such a basis, as was proved by Enflo in [Enfl73]. However, all such spaces have a Markushevich basis (from now on called an M-basis), a result due to Markushevich himself that elaborates on the basic Gram-Schmidt orthogonal process. It will be proved in Chapter 5 that many nonseparable Banach spaces also possess M-bases, even with some extra features, allowing actual computations and opening a way to classification of Banach spaces.