Completability of a uniform space, metric space and topological vector space?

Question

From Wikipedia

In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. (This implies that every Hausdorff topological vector space is completely regular.[2]) The vector space operations of addition and scalar multiplication are actually uniformly continuous. Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space.

1. I was wondering generally what some conditions are for a uniform space to be completed? I couldn't find relevant information on Wikipedia and likes.
2. Similar question for a metric space?
3. Because of what, can every topological vector space be completed?

Thanks and regards!

Answer 1

1. Every uniform space has a unique (up to uniform isomorphism) completion.

2. Every metric space can be completed (this is a standard result in analysis).

The book General Topology by Engelking has a whole section on uniform spaces if you are interested in seeing the proofs.

Answer 2

Let $\hat{X}$ be the completion of the TVS $(X,a,s)$ as a uniform space (as in the mentioned General Topology books). Herein $a$ and $s$ denote the vector space addition and scalar multiplication. As $a$ and $s$ are uniformly continuous maps, they have a unique uniformly continuous extension $\hat{a}$ and $\hat{s}$ from $\hat{X} \times \hat{X}$ resp. $K \times \hat{X}$ to $\hat{X}$ where $K$ denotes the underlying field.

Now you can easily verify that $\hat{a}$ and $\hat{s}$ fulfill the vector space axioms on $\hat{X}$. This is the key argument: The purely non-algebraic uniform completion of a TVS preserves the algebraic vector space structure and turns the uniform completion into a complete TVS.