ABSTRACT
Underlying functional analysis is the specification of a set of objects V on which is imposed some additional structure, typically representing generalizations of notions natural to multidimensional Euclidean vector geometry. The objects in V may be functions, infinite dimensional sequences or random variables, in addition to Euclidean vectors, but may all be conveniently referred to simply as vectors. We therefore need to define abstract notions of vector algebra, as well as quantitative abstractions of concepts such as the distance between vectors, the length of a vector, and the angle between vectors. There is a hierarchical progression to the definition of this structure, and applications will differ in the amount of structure required.
