The spaces C̥, ¬ı, and ¬.

Abstract

Given the sets C0, all sequences of real numbers converging to zero, l1 , all absolutely summable sequences of real numbers, and 1∞, all bounded sequences of real numbers, the study of infinite dimensional vector spaces is developed. The use of basic analysis concepts allows for the proofs that l1 is a subset of C0, and that C0 is a subset of l∞. The definitions and theorems of vector spaces allow the proofs that each of these spaces are vector spaces and have norms defined on them. Linear mappings among these spaces and from one to the set of real numbers are discussed as well as the norm of such functionals. Again using analysis, the concept of continuous functionals is developed. With this knowledge the topic of a dual space, or the space of all bounded linear functionals on a normed linear space, is investigated. Finally, with the introduction of complete spaces it is concluded that C0, l1, and 1∞ are all Banach spaces. This result leads to the consideration of extreme points, unconditional convergence, the Dvoretsky-Rogers Theorem and the Hahn-Banach Theorem.