Given any linear space X, it follows from the existence of a Hamel basis for X and the fact that any linear functional is determined by its values on the Hamel basis, that the algebraic dual X# is generally a “substantial” space. We know, from Remark 4.10.2, that for an infinite dimensional normed linear space (X, ∥·∥), the dual X* is a proper linear subspace of X#.

For the development of a theory of normed linear spaces in general, quite apart from particular examples or classes of examples, it is important to know that given any normed linear space (X, ∥·∥), its dual X* is also “substantial enough” and by this we mean that we have a dual which generalises sufficiently the properties we are accustomed to associate with the dual of a Euclidean space or indeed, with the duals of the familiar example spaces.

We now use the Axiom of Choice in the form of Zorn's Lemma, (see Appendix A. 1), to prove the Hahn–Banach Theorem, an existence theorem which is crucial for the development of our general theory. The theorem assures us that for any nontrivial normed linear space there is always an adequate supply of continuous linear functionals.

The immediate application of this result is in the study of the structure of the second dual X** of a normed linear space (X, ∥·∥) and of the relation between the space X and its duals X* and X**.