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Erschienen in:
2013 | OriginalPaper | Buchkapitel
1. Normed Spaces
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
We now set off on an expedition through the vast subject of functional analysis. No doubt the reader has some familiarity with this place, and will recognize some of the early landmarks of the journey. Our starting point is the study of normed spaces, which are situated at the confluence of two far-reaching mathematical abstractions: vector spaces, and topology. The setting is that of a vector space over the real numbers \({\mathbb{R}}\). The central idea of this chapter is that of a norm. It is studied in detail, along with the attendant concept of linear operators. The dual space is introduced. The chapter ends with a presentation of derivates, directional derivatives, tangent and normal vectors. These constructs, which allow one to reduce nonlinear situations to linear ones, will play a central role in later developments.