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How does one show this?

I could use a hint.

Thanks

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    What is the source of this problem? What have you tried so far?2012-02-25
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    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.2012-02-25
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    @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".2012-02-25
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    What is a 'metric linear space'?2012-02-25

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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).