In the book Sadri Hassani, the author gave an example of a Cauchy series in which the vector at infinity does not lie in the original space. The example is shown below:-
He then goes on to define that the Hilbert space of square-integrable function is complete and isomorphic to Lebesgue spaces. However, he doesn't prove what you need in order to complete a vector space. For example, to complete rational numbers, you need real numbers.
My question is, given a vector space, is there a way to complete it in general. It would be extremely helpful if you can take the example mentioned and complete it if possible?