15 - We Go Deeper, DeeperWe Go (into the Structure of Complete Spaces)

Summary

The celebrated Baire’s category theorem says that a complete space cannot be represented as a countable union of nowhere dense sets. This is a fundamental description of the structure of complete spaces. Because of this, it is fitting to derive the Banach–Steinhaus theorem as a consequence of Baire’s. This is what we do at the beginning of this chapter. We also show that the set of differentiable functions is quite small (i.e. meagre) in the space of continuous functions. As further consequences of Baire’s theorem we discuss two other fundamental results of functional analysis – the open mapping theorem and the closed graph theorem – together with some of their most immediate applications. In the meantime, we use the Banach–Steinhaus theorem to show that a Fourier series cannot converge uniformly for all continuous (and periodic) functions.