Duality and the Hahn—Banach Theorem

In this chapter we return to the study of linear functionals and dual spaces, which were defined and briefly considered in Section 4.3. In the first section we obtain specific representations of the duals of some particular spaces. We then give various formulations of the so called Hahn—Banach theorem. This theorem enables us to construct linear functionals, on general spaces, with specific properties. This then enables us to derive various general properties of dual spaces, and also of second dual spaces. We conclude the chapter by considering “projection operators” and “weak convergence” of sequences — these topics have many uses, and rely on some of the results established earlier in the chapter.