Niveau: Supérieur, Master
Chapter 5 Banach spaces Analyse Master 1 Cours de Francis Clarke (2011) 5.1 Completeness of normed spaces A normed space X is said to be a Banach space if its metric topology is complete: every Cauchy sequence xi in X (that is, one that satisfies lim i, j?∞ xi ? x j = 0) admits a limit in X : there exists a point x ? X such that xi? x? 0. Informally, the absence of such a point x would mean 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. Consider, for example, the vector space Q of rational numbers. The minimization of the function (x2?2)2 does not admit a solution over Q, as the Greek mathematicians of antiquity were able to prove. The existence of solutions to minimization problems is not the only compelling reason to require the completeness of a normed space, as we shall see. The property is essential for such tools as uniform boundedness, variational principles, and weak compactness. The reader is asked to observe that completeness of a normed space is invariant under equivalent norms, and that a closed subspace of a Banach space is itself a Banach space. It is easy to see as well that the Cartesian product of two Banach spaces is a Banach space.
- since acp
- minimization principles
- lim n?∞
- banach spaces
- banach space
- zn ?
- theorem readily
- every cauchy sequence