Practical implications of a vector space being a topological vector space

Question

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?

Answer 1

By having both the vector space and topological structures interacting, you will have the entire theory of topological vector spaces at hand. This is more than just the theory of vector spaces and theory of topological spaces acting independently.

One thing, for example, is that they are uniform spaces. This means essentially that the neighborhoods of any two points look the same.

Another thing is that even if you only require the separation axiom $T_0$, because of the way the vector space interacts with the topology, the topology must actually even be $T_{3\frac{1}{2}}$! If it's a finite dimensional vector space, you also have that $V$ is locally compact.

Another thing is that the continuous space of linear functionals $V^\ast$ gives rise to another interesting topology on $V$.