In this chapter, we consider spaces of differentiable mappings as infinite-dimensional spaces. These spaces will then serve as the model spaces for manifolds of mappings, i.e. manifolds of differentiable mappings between manifolds. The resulting manifolds will allow us to construct essential examples in later chapters, such as the diffeomorphism groups. Moreover, they arise naturally in the context of many applications such as shape analysis and the geometric treatment of partial differential equations which will be discussed later. Finally, we introduce an indispensable tool for the treatment of differentiable mappings on manifolds of mappings in this chapter: the exponential law. roughly speaking, the exponential law allows to interpret a smooth map taking values in a manifold of mappings as a smooth map of two arguments. Thus differentiability questions can often be relegated to a finite dimenisonal setting.

This short appendix covers several basic concepts of topology, which we review for the readers convenience. For example, the compact open topology on spaces of continuous functions is discussed together with an appropriate version of the exponential law.