The purpose of this paper is threefold. First, we wish to make clear to the reader the structural similarities and differences between finite- and infinite-dimensional vector spaces. In doing so, the reader with some knowledge of linear algebra and real or complex analysis should be comfortable with the generalizations and abstractions contained below. Second, we wish to explore in detail the mathematical ideas and techniques at the foundation of functional analysis. In other words, we shall develop a context of study and, in doing so, convey to the reader some sense of what one actually means by functional analysis. The proofs are often elegant, and the techniques used to solve problems are quite easily applied to other areas of functional analysis. These two points shall be the topic of the first three chapters. Third, we wish to impress upon the reader the power of functional analysis to solve pure and applied problems in other areas of mathematics. This will be done especially in Chapter 4, where we apply a very general fixed point theorem to a particular partial differential equation to establish existence and smoothness of generalized solutions. This latter work was completed during the author's NSF-funded internship at Cornell University and credit should be also attributed to Phillip Whitman, Frances Hammock, and Alexander Meadows.

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