ABSTRACT
In this chapter, we will give a brief introduction to analysis-questions of convergence, continuity, and so forth-in vector spaces. Our treatment is necessarily brief; we wish to make two basic points. First of all, all norms on a given finite-dimensional vector space are equivalent, in the sense that they define the same notion of convergence. We learned early in this book that all finite-dimensional vector spaces are isomorphic in the algebraic sense; the results of this chapter (Section 10.1) show that this is true in the analytic sense as well. Second, infinite-dimensional vector spaces are much more complicated. In particular, it is possible to define nonequivalent norms on an infinite-dimensional vector spaces. We will find it useful to define notions of convergence that do not depend on a norm.
