Octonionic Functional Analysis And Non-associative Categories

This paper studies octonion functional analysis and non-associative category.It is a great challenge to generalize functional analysis from associative to non-associative cases.The octonion Hilbert space was given an axiomatic definition as early as 1964 by Goldstine and Horwitz.But one of the axioms as we recently discovered is not independent,this redundant axiom hampered the development of the theory of octonion Hilbert spaces,causing the development of the theory to almost come to a standstill.Since then,research on octonion Hilbert space theory has focused on special octonion Hilbert spaces with tensor decomposition,focusing on real linear or octonion linear operators.This paper discovers for the first time that the research object of octonion functional analysis is para-linearity.This means that para-linearity is a substitute for linearity in the non-associative case.From the category point of view,the object of octonion functional analysis is the octonion Hilbert space,and its morphisms are para-linear operators.This category generalizes the classical category theory from the associative case to the nonassociative case.We generalize the classical functional analysis to the octonion case.We encountered many obstacles along the way:1.The submodule structure of the octonion case is very complicated.For example,a submodule generated by an element x no longer has the form Ox;2.Different from the classical case,the definition of the para-linear mapping of octonions requires the help of the associators and the real part operator;3.Introducing the octonion module structure into the real linear space of the octonion para-linear mappings requires defining a new octonion scalar multiplication.Since the para-linear mapping cannot preserve the para-linearity under ordinary composition,this leads us to introduce regular composition;4.An octonion para-linear operator is a highly nonlinear operator,so the kernel and range of a para-linear operator are no longer submodules,which makes it more difficult to establish the theory of para-linear operator of octonion.Our main results are as follows:1.For the first time,we define Hilbert spaces on divisible algebras,and establish the octonion Riesz representation theorem for para-linear functionals.2.It is well known that Parseval’s equation does not hold in the case of octonions.This hinders the application of Parseval’s equation.In this paper,the criterion for the Parseval’s equation being valid are given,that is,the basis in consideration is a weak associative basis.3.We extend the bilinear form to the octonion case so that the concept of paraconjugate bilinear functional is introduced and the Lax Milgram theorem is established.4.We give the necessary and sufficient conditions for octonion self-adjoint operators to be characterized by quadratic form.Unlike the classical case,the quadratic form induced by the octonion self-adjoint operator is generally not always real-valued.5.We establish the octonion Hahn-Banach theorem and the octonion BanachAlaoglu theorem.6.We initiate the study of non-associative categories and establish the Yoneda lemma for the non-associative category.