Octonionic Hahn–Banach theorem for para-linear functionals

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 $x\in X$ is no longer of the form $O\left\{x\right\}$. 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 $O$-modules.