Note on the Hahn-Banach Theorem in a Partially Ordered Vector Space

The Hahn-Banach theorem is one of the most fundamental theorems in the functional analysis theory. This theorem is well known in the case where the range space is the real number system as follows. Let p be a sublinear mapping from a vector space X into the real number system R, Y a subspace of X, and q a linear mapping from Y into R such that q ≤ p on Y. Then there exists a linear mapping g fromX into R such that g = q on Y and g ≤ p onX. It is known that this theorem is established in the case where the range space is a Dedekind complete Riesz space as follows [1–3]. Let p be a sublinear mapping from a vector space X into a Dedekind complete Riesz space E, Y a subspace ofX and q a linear mapping from Y into E such that q ≤ p on Y. Then there exists a linear mapping g fromX into E such that g = q on Y and g ≤ p onX. On the other hand, Hirano et al. [4] showed the HahnBanach theorem by using the Markov-Kakutani fixed point theorem [5] in the case where the range space is the real number system. In this paper, motivated by Hirano et al. [4], we give a proof of the Hahn-Banach theorem using a fixed point theorem. We show the Hahn-Banach theorem in the case where the range space is a Dedekind complete partially ordered vector space (Theorem 10). Moreover, we show the Mazur-Orlicz theorem in a Dedekind complete partially ordered vector space (Theorem 11).