Definition 2

For every right A-module E, the right A-module $\operatorname{Hom}_A(E, A_s)$ is called the dual module of E (or simply the dual† of E) and its elements are called the linear forms on E.

If E is a right A-module, the set $\operatorname{Hom}_A(E, A_d)$ with its canonical left A-module structure is likewise called the dual of E and its elements are called linear forms on E.

† In Topological Vector Spaces, IV, we shall define, for vector spaces with a topology, a notion of "dual space" which will depend on this topology and will be distinct from the one defined here. The reader must beware of incautiously applying to the "topological" dual space the properties of the "algebraic" dual which are established in this paragraph.

In this chapter, $E^*$ will be used to denote the dual of a (left or right) A-module $E$.