What is meant by "functional analysis is the study of vector spaces endowed with a topology"

Question

Lecture notes on Functional Analysis by Razvan Gelca open with the definition:

Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces.

But Topology is the study of sets. I am taking it right now, MATH300 using "Topology" by Munkres.

What does Topology mean in the context of the above quote?

Answer 1

First of all, topology is not the study of sets. Set theory is the study of sets. Topology is the study of sets endowed with an additional structure, called topology, and you will find all the details in your book.

Functional analysis is a broad subject nowadays, but it can be described as the study of topological vector spaces and linear operators acting on them. What are topological vector spaces? As somebody writes in his/her answer, it is the study of those topologies that are in some sense compatible with the two operations that we define on vector spaces: the sum of two vectors and the multiplication of a vector by a number. These two operations must be continuous in order to speak of a topological vector spaces. You will discover that normed spaces are topological vector spaces, for instance. So functional analysis is not "$+$ and $\times$" as you commented: it is the marriage of $+$ and $\times$ with a continuity requirement.

Answer 2

A vector space is a set, so it can have a topology. In this case, we are usually interested in topologies which are 'compatible' with the vector space structure. Specifically, the topology should be such that addition and scalar multiplication are continuous maps.