Extensions of functionals on octonionic linear spaces

Abstract

The Hahn--Banach theorem states that any linear functional defined on a linear subspace of a normed linear space has a norm-preserving extension to the whole space. Until 1938 it was not known whether or not this result was restricted to real linear spaces and real-valued functionals. In that year Sobczyk and Bohnenblust [3] proved that a complex linear functional defined on a complex linear subspace of a complex normed linear space has a norm-preserving extension to the entire space. Moreover, they showed that the subspace must be complex linear (not merely real linear) in order for such an extension to necessarily exist. In that same year Suhomlinov [4] generalized to prove the analogue of the Hahn--Banach theorem for quaternionic normed linear spaces as well. This paper investigates the extent to which the Hahn---Banach theorem is valid in the still more general case of linear spaces which are defined over the non-associative algebra O of real octonions. Real octonions are all hypercomplex numbers of the tbrm a o + a l e l + a 2 e ~ + +azez+a4e4+ane~+ane6+aTe7 where a0, a~, ..., ar are real coefficients with distributive products defined by the following multiplication rules for the unit 1 and the seven basic imaginary units ea, e 2, ..., er: