The Hahn-Banach Theorem allows for the extension of linear maps defined on subspaces of normed spaces to the whole space in a way that respects sublinear bounds. The simplest case is the extension of bounded linear functionals from subspaces to the whole space. We prove this result here, first for real spaces and then for complex spaces. The proof requires use of Zorn’s Lemma, unless we assume that the underlying space is separable.

We give some important applications of the Hahn-Banach Theorem. We prove the existence of a support functional and hence that X^* separates points in X. Then we prove the existence of a functional that encodes the distance from a linear subspace, which is an important ingredient in a number of subsequent proofs. We show that separability of X^* implies separability of X, define the Banach adjoint of a linear map (between Banach spaces), and prove the existence of ‘generalised Banach limits’.

Basic topics on Banach space theory needed for the text are reviewed. Hahn-Banach theorem, Baire’s theorem, uniform boundedness principle, closed graph theorem, weak topologies, Banach-Alaoglu theorem, unconditional basis, Banach sequence spaces, summing operators, factorable operators, cotype, Kahane inequality.