Banach spaces

Abstract

A normed space X is said to be a Banach space if its metric topology is complete. This means that every Cauchy sequence x _i in X admits a limit in X: there exists a point x∈ X such that ∥ x _i −x ∥→ 0. Informally, the reader may understand the absence of such a point x as meaning that the space has a hole where x should be. For purposes of minimization, one of our principal themes, it is clear that the existence of minimizers is imperiled by such voids. The existence of solutions to minimization problems is not the only compelling reason, however, to require the completeness of a normed space, as we shall see. The property is essential in making available to us certain basic tools developed in this chapter, such as uniform boundedness, minimization principles, and weak compactness.