Octonionic Hahn–Banach theorem for para‐linear functionals

Abstract Goldstine and Horwitz introduced the octonionic Hilbert space in 1964, sparking extensive research into octonionic linear operators. A recent discovery unveiled a significant characteristic of octonionic Hilbert spaces: an axiom, once deemed insurmountable, has been found not to be independent of other axioms. This discovery gives rise to a novel concept known as octonionic para‐linear functionals. The purpose of this paper is to establish the octonionic Hahn–Banach theorem for para‐linear functionals. Due to the non‐associativity, the submodule generalized by an element is no longer of the form . This phenomenon makes it difficult to apply the octonionic Hahn–Banach theorem if the theorem holds only for functionals defined on octonionic submodules. In this paper, we introduce a notion of para‐linear functionals on real subspaces and establish the octonionic Hahn–Banach theorem for such functionals. Then we apply it to obtain some valuable corollaries and discuss the reflexivity of Banach ‐modules.