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Akademische Abhandlung--Helsinki. / Includes bibliographical references (p. [38]).

Thesis--Auburn University, 2009. / Abstract. Vita. Includes bibliographical references (p. 23).

Several properties of Hausdorff continua are considered in this paper. However, the major emphasis is on developing the properties of the hyperspaces 2x and C(X) of a Hausdorff continuum X. Preliminary definitions and notation are introduced in Chapter I. Chapters II and III deal with the topological structure of the hyperspaces and the concept of topological convergence. Properties of 2x and C(X) are investigated in Chapter IV, while Chapters V and VI are devoted to the Hausdorff continuum X. Chapter VII consists of theorems pertaining to Whitney maps and order arcs in 2x. Examples of C(X) are provided in Chapter VIII. Inverse sequences of Hausdorff continua and of their hyperspaces are considered in Chapter IX.

In this thesis two problems concerning linear transformations are discussed. Both problems involve linear transformations which are not in some sense semi-simple; otherwise they are unrelated. In part I we present a proof of the theorem that a linear transformation, of a finite dimensional vector space over a field, which has the property that the irreducible factors of its minimal polynomial are separable is the sum of a semi-simple linear transformation and a nilpotent linear transformation, which commute with the original transformation and are polynomials in the original transformation.. We present an example to show that such a decomposition is not always possible. In parts II and III we obtain some representation theorems for closed algebras of linear transformations on a Banach space which are generated by spectral operators. Since such an algebra is the direct sum of its radical and a space of continuous functions its radical can be investigated more readily than the radical of an arbitrary non-semi- simple commutative Banach algebra. In part II we remark that the reduction theory for rings of operators allows one to reduce the problem of representing a spectral operator, T, on a Hilbert space to the problem of representing a quasi-nilpotent operator. When T is of type m+1 and has a "simple" spectrum it is quite easy to obtain an explicit representation of T. In part III we consider spectral operators on a Banach space. We impose quite stringent conditions, hoping that the theorems obtained for these special cases will serve as a model for more general theorems. The knowledge obtained at least delimits the possibilities. We assume that T is of type m+1 and has a "simple" spectrum. One other condition, which is satisfied if the space, X, on which T acts is separable, is imposed. We are then able to obtain a representation of X as a function space. These function spaces are modelled on the analogy of the Orlicz spaces. We are also able to obtain a representation of the not necessarily semi-simple algebra generated by T and its associated projections as an algebra of functions. / Science, Faculty of / Mathematics, Department of / Graduate

Dissertation (Ph.D.)--Auburn University, / Abstract. Vita. Includes bibliographic references (p.88).

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 99-116).

No description available.

Call number: LD2668 .T4 1985 T36 / Master of Science

This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.