This paper is a study of linear operators defined on normed linear spaces. A basic knowledge of set theory and vector spaces is assumed, and all spaces considered have real vector spaces. The first chapter is a general introduction that contains assumed definitions and theorems. Included in this chapter is material concerning linear functionals, continuity, and boundedness. The second chapter contains the proofs of three fundamental theorems of linear analysis: the Open Mapping Theorem, the Hahn-Banach Theorem, and the Uniform Boundedness Principle. The third chapter is concerned with applying some of the results established in earlier chapters. In particular, the concepts of compact operators and Schauder bases are introduced, and a proof that an operator is compact if and only if its adjoint is compact is included. This chapter concludes with a proof of an important application of the Open Mapping Theorem, namely, the Closed Graph Theorem.

The purpose of this paper is to investigate the nature and practical uses of Chebyshev polynomials. Chapter I gives recognition to mathematicians responsible for studies in this area. Chapter II enumerates several mathematical situations in which the polynomials naturally arise and suggests reasons for the pursuance of their study. Chapter III includes: Chebyshev polynomials as related to "best" polynomial approximation, Chebyshev series, and methods of producing polynomial approximations to continuous functions. Chapter IV discusses the use of Chebyshev polynomials to solve certain differential equations and Chebyshev-Gauss quadrature.

The purpose of this thesis is to study the equivalence relation between sets A and B: A o B if and only if there exists a one to one function f from A onto B. In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied and three different proofs of it are given. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and some fundamental theorems on cardinal arithmetic are proved.