In this work certain aspects of Functional Analysis are considered in the setting of linear spaces over the division rings of the real Quaternions and the real Cayley algebra. The basic structure of Banach spaces over these division rings and the rings of bounded operators on these spaces is developed. Examples of finite and infinite dimensional spaces over these division rings are given. Questions concerning linear functionals, the Hahn-Banach Theorem and Reflexivity are considered. The Stone-Weierstrass Theorem is proven for functions with values in a real Cayley Dickson algebra of dimension n. The concepts of inner product spaces and Hilbert spaces over the Quaternions and the Cayley algebra are developed. An extensive study of Hilbert spaces over the Quaternions is carried out. In the case of Hilbert spaces over the Quaternions, the Riesz-Representation Theorem and the Jordan-von Neumann Theorem are proven. In addition, spectral theorems for both self-adjoint and normal operators are proven for finite dimensional Hilbert spaces. These results are extended to infinite dimensional spaces for the cases of compact self-adjoint operators and compact normal operators. The spectrum of an arbitrary bounded Hermitian operator on a Hilbert space over the Quaternions is shown to be non-void. A generalization of the Fourier Transform for functions in L[1 over Q](-infinity, infinity) and L[2 over Q]( ](-infinity, infinity) is given. The Plancherel Theorem is proven for functions in L[2 over Q](-infinity, infinity). Finally, the Jordan-von Neumann theorem is proven for a Hilbert space over the Cayley algebra --Abstract, pages ii-iii
