A Generalization of the Hahn-banach Theorem

was established by Hahn [1; p. 217] in 1927, and independently by Banach [2; p. 212] in 1929, who also generalized Theorem 0 for real spaces, to the situation in which the functional q :E^>R is an arbitrary subadditive, positive homogeneous functional [2; p. 226]. Theorem 0 was not established for complex spaces until 1938, when it was deduced from the real theorem by Bohnenblust and Sobczyk [3], and by Soukhomlinoff [4], who also deduced the analogue for vector spaces over the quaternions. In this paper, we generalize Theorem 0 to modules over a Banach algebra A: the proof yields in particular a variation of the classical proof of the complex theorem, and the theorem for quaternions. Our generalization is related to the extension theorem of Bonsall and Goldie [5; p. 13], and interacts fruitfully with the theorem established by Nachbin [6; p. 30].