what is the meaning of a power set of topological vector space?

Question

Given a topological vector space, what is the power set of this space meaning?

thanks a lot.

and I really appreciate if a straightforward and simple explanation of topological vector space is illustrated too.

Answer 1

The power set of a set $X$ is the set of subsets of $X$, often denoted $2^X$. This doesn't really have anything to do with topological vector spaces per se, except inasmuch as a topological vector space is a set.

A topological vector space is a vector space equipped with a topology that makes the addition and scalar multiplication operations continuous. (In order to make sense of the statement "scalar multiplication is continuous", the underlying field needs to have a topology as well. For example, the field could be $\mathbb R$ or $\mathbb C$, equipped with the usual topologies.)

As a motivating example, $\mathbb R^n$ is a vector space, and it also has a topology on it (the Euclidean topology) with respect to which vector addition and scalar multiplication are continuous maps. That is, the function $\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ defined by $(x,y)\mapsto x+y$ and the function $\mathbb R\times\mathbb R^n\to\mathbb R^n$ defined by $(\lambda,x)\mapsto \lambda x$ are both continuous. So $\mathbb R^n$ is a topological vector space.