Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call
themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both.