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Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.

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  • $\begingroup$ By the very definition of subspace, yes. $\endgroup$ – Ataraxia Aug 18 '13 at 18:19
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    $\begingroup$ Finite dimensional normed spaces of the same dimension are isomorphic. A finite subspace of a normed vector space $X$ is thus isomorphic to some $\ell_2^n$. As such, it is complete; thus closed. $\endgroup$ – David Mitra Aug 18 '13 at 18:29
  • $\begingroup$ @DavidMitra Thanks $\endgroup$ – chandu1729 Aug 18 '13 at 18:30
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    $\begingroup$ Perhas chandu assumes real scalars? Or at least a complete field for the scalars... otherwise counterexamples exist. $\endgroup$ – GEdgar Aug 18 '13 at 19:02
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    $\begingroup$ Topologically closed $\endgroup$ – chandu1729 Aug 18 '13 at 19:36
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Finite dimensional normed spaces of the same dimension are isomorphic. A finite subspace of a normed vector space X is thus isomorphic to some $\ell_2^n$. As such, it is complete; thus closed.

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