In this chapter we explore some simple properties of functionals defined on a normed linear space. We indicate how linear functionals can be extended from a subspace to the entire normed linear space and this makes the normed linear space richer by new sets of linear functionals. The stage is thus set for an adequate theory of conjugate spaces, which is an essential part of the general theory of normed linear spaces. The Hahn-Banach extension theorem plays a pivotal role in extending linear functionals from a subspace to an entire normed linear space. The theorem was discovered by H. Hahn (1927) [23], rediscovered in its present, more general form (5.2.2) by S. Banach (1929) [5]. The theorem was further generalized to complex spaces (5.1.8) by H.F. Bohnenblust and A. Sobezyk (1938) [8].

Besides the Hahn-Banach extension theorem, there is another important theorem discovered by Hahn-Banach which is known as Hahn-Banach separation theorem. While the Hahn-Banach extension theorem is analytic in nature, the Hahn-Banach separation theorem is geometric in nature.