Introduction

The goal of this appendix is to introduce several types of vector spaces and explain how they are related. In the end, most of the vector spaces we work with – which are built out of smooth or distributional sections of vector bundles – behave nicely in whichever framework one chooses to use, but it is important to have a setting where abstract constructions behave well. In particular, we will do homological algebra with infinite-dimensional vector spaces, and that requires care. Below, we introduce the underlying “functional analysis” that we need (i.e., we describe here just the vector spaces and discuss the homological issues in a separate appendix). For a briefer overview, see Section 3.5 in Chapter 3.

There are four main categories of vector spaces that we care about:

1. • LCTVS, the category of locally convex Hausdorff topological vector spaces

2. • BVS, the category of bornological vector spaces

3. • CVS, the category of convenient vector spaces

4. • DVS, the category of differentiable vector spaces

The first three categories are vector spaces equipped with some extra structure, like a topology or bornology, satisfying some list of properties. The category DVS consists of sheaves of vector spaces on the site of smooth manifolds, equipped with some extra structure that allows us to differentiate sections (hence the name).

The main idea is that DVS provides a natural place to compare and relate vector spaces that arise in differential geometry and physics. As a summary of the relationships between these categories, we have the following diagram of All the functors into DVS preserve limits. The functors and out of BVS are left adjoints. The functors into DVS all factor as a right adjoint followed by the functor difβ. For example, dift : LCTVS → DVS is the composition difβ °born, where the bornologification functor born : LCTVS → BVS is the right adjoint to the inclusion incβ : BVS → LCTVS.

For our work, it is also important to understand multilinear maps between vector spaces (in the categories mentioned earlier). Of course, it is pleasant to have a tensor product that represents bilinear maps. In the infinite-dimensional setting, however, the tensor product is far more complicated than in the finitedimensional setting, with many different versions of the tensor product, each possessing various virtues and defects.