Doctoral Dissertations


"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.


Penico, Anthony J., 1923-2011

Committee Member(s)

Hicks, Troy L.
Pagano, Sylvester J., 1924-2006
Falcao, Linda M.
Brown, Harry A., 1925-1995
Stanojević, Časlav V., 1928-2008


Mathematics and Statistics

Degree Name

Ph. D. in Mathematics


University of Missouri--Rolla

Publication Date



vii, 171 pages

Note about bibliography

Includes bibliographical references (pages 163-170).


© 1970 James Edward Jamison, All rights reserved.

Document Type

Dissertation - Open Access

File Type




Subject Headings

Function spaces
Banach spaces
Isometrics (Mathematics)
Cayley algebras

Thesis Number

T 2372

Print OCLC #


Electronic OCLC #


Included in

Mathematics Commons