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How does one show this?

I could use a hint.


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What is the source of this problem? What have you tried so far? – Jesse Madnick Feb 25 '12 at 16:04
Some of this terminology is somewhat non-standard. I assume you're asking how to show that a vector subspace of a finite-dimensional normed vector space is closed. – Jesse Madnick Feb 25 '12 at 16:04
@Jesse: in some circles, at least some Functional Analysis ones, the term "subspace" is reserved for the closed ones; for not necessarily ones, the term "linear manifold" is used. And if you move it from the origin (by adding a fixed vector) you get an "affine manifold". – Martin Argerami Feb 25 '12 at 16:47
What is a 'metric linear space'? – wildildildlife Feb 25 '12 at 17:39
Thanks for the commments. – user25720 Feb 26 '12 at 15:29
up vote 1 down vote accepted

My hint: move your problem to the origin. Then you will have a subspace (or linear manifold), and you can prove that convergence is exactly convergence in coordinates (this is where finite-dimensionality enters into play).

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