Let A be a topological algebra. A (Schauder) basis {xn} in A is called an orthogonal basis if xnxm = δnmxn, n,m ε N (δ denotes Kronecker's delta). A basis in A of the form {zⁿ : n=0,1,...}, z ɛ A, is called a cyclic basis. This thesis is concerned with the structure of topological algebras possesing bases of these types. It is shown how the existence of such bases determines algebraic and topological properties of A. An interesting connection between the dense maximal ideals and the topological dual of certain types of topological algebras having unconditional orthogonal bases is explored. These results are used to obtain characterizations of some important F-algebras in terms of the type of bases they possess.

Doctor of Philosophy (PhD)