Conclusion:
While at first, with no background in the field of functional analysis, Banach Spaces seem daunting, complex, relatively useless, and unnecessarily confusing, this is far from the case. Banach Spaces are useful on their own in different kinds of algebras (fields). However, Banach Spaces and applications of them have allowed the field of functional analysis to initially exist and eventually flourish. Before reading about these, I had never heard of functional analysis. However, after learning more about the field, its applications are ubiquitous in the world around us. From applications in economics to quantum mechanics, functional analysis is an extremely broad field that encompasses so much and is used in so many places. For example, the Von Neumann model was one of the earliest models of equilibrium in economics. Through the use of functional analysis, Von Neumann was able to accurately simulate the concept of economic equilibrium in a market. The Von Neumann model was, “ a brilliant mathematical synthesis of the classical ideas concerned with the proportions required by economic equilibrium. It was on the other hand a forerunner of modern mathematical economics, which became fully developed only some decades later, under the influence of neoclassical economics” (http://unipub.lib.uni-corvinus.hu/245/1/wp_2003_5_zalai.pdf). Thus, although it is a relatively obscure field, functional analysis and Banach Spaces have far reaching influence across many aspects of our modern day understanding of how the world works. The information provided in the above essay is a fraction of the use and complexity of Banach Spaces but serves as a decent foundation for a general understanding of the pieces and conditions required for a Banach Space to exist and some of their uses in the field of functional analysis.
