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.
KeywordsBanach Space Basic Sequence Separable Banach Space Minimal System Biorthogonal System
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