I am having a hard time working out the completion of certain spaces. But first I'd like to be able to fully understand what exactly the completion of a space is. Here is the way I see it. (It could be wrong, or unclear) If you have a normed vector space V [Does the space you want to complete have to be normed?] then there is a Banach space X and a linear isometric isomorphism T of V onto T(V), where T(V) is a dense subspace of X. Ok, so you have a normed space V, which does not have to be complete, however it can be. [If V were complete, does this mean we could simply map, via the linear isometric isomorphism T, straight onto X?] Then we have a linear isometric isomorphism T, which maps elements in V to T(V). But T(V) is no ordinary space, it is actually a dense subspace of a, perhaps, larger Banach space X. Whats more, the isomorphism which does this, T, and the resulting Banach space X is actually unique. [So for every V there is a unique map T and Banach space X?]. Now is this Banach space, X the completion of V? Also, I don't seem to have any easy, illustrative examples of the completion of spaces. If anyone could be so kind as to provide a example or two, or has any discussion on this topic, or has an illuminating description which might help me understand this better, it would be much appreciated.