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Banach Spaces and Differential Calculus
Manifolds have enough structure to allow differentiation of maps between them. To set the stage for these
concepts requires a development of differential calculus in linear spaces from a geometric point of view. The
goal of this chapter is to provide this perspective.
Perhaps the most important theorem for later use is the Implicit Function Theorem. A fairlydetailed
exposition of this topic will be given with examples appropriate for use in manifold theory. The basic
language of tangents, the derivative as a linear map, and the chain rule, while elementary, are important for
developing geometric and analytic skills needed in manifold theory.
The main goal is to develop the theoryof finite-dimensional manifolds. However, it is instructive and effi-
cient to do the infinite-dimensional avoid being sidetracked byinfinite-dimensional
technicalities at this stage, some functional analysis background and other topics special to the infinite-
dimensional case are presented in supplements. With this arrangement readers who wish to concentrate on
the finite-dimensional theorycan do so with a minimum of distraction.
Banach Spaces
It is assumed the reader is familiar with the concept of a real plex vector space. Banach spaces are
vector spaces with the additional structure of a norm that defines plete metric space. While most of
this book is concerned with finite-dimensional spaces, much of the theoryis reallyno harder in the general
case, and the infinite-dimensional case is needed for certain applications. Thus, it makes sense to work in the
setting of Banach spaces. In addition, although the primaryconcern is with real Banach spaces, the basic
concepts needed plex Banach spaces are introduced with little extra effort.
Normed Spaces. We begin with the notion of a normed space; that is, a space in which one has a length
measure for vectors.
Definition. A norm on a real (complex) vec