Introduction:

In this essay, we will be discussing Banach Spaces and their importance to the field of functional analysis, which deals with the transformation of functions and their properties. A Banach Space, as defined by professors Todd Rowland and Mohammad Sal Moslehian is “a complete vector space B with a norm (||.||)”. Norms will be discussed first and are essentially functions that map a vector space and map it as a function of scalars. The definition of a complete vector space will then be discussed as the coupling of a vector space and a norm. Banach spaces can only begin to be discussed upon understanding Cauchy sequences which are sequences where the difference between their elements converges to an arbitrary value. Lastly, with all of these puzzle pieces at our disposal, Banach Spaces can finally be discussed in the final section.

Banach spaces first entered the field of mathematics in a meaningful way in the 1920s and 1930s when the work of Polish mathematician Stefan Banach came together with the work of German Mathematician David Hilbert. The combination of their work arguably started the field of functional analysis. As Banach Spaces are a solidly defined object within mathematics, not much has changed regarding their definition over the years. However, their applications in the field of functional analysis are practically endless and continue to expand.