- #1
Oxymoron
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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.
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.