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How does it follow that a subset of a normed vector space cannot be open if it does not contain an open ball $B_{\epsilon}(0)$ where $\epsilon > 0$?

I just want to confirm also that for normed vector spaces of any dimension there are no open subspaces? For finite dimensional spaces all subspaces are closed. Subspaces of Hilbert spaces are complete iff they are closed.

Thanks for help

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    $\begingroup$ A subset can be open without containing a ball around $0$. A subspace, in the sense of linear subspace, always contains $0$, and if it is open, it therefore contains a ball around $0$. And therefore it is then the entire space. $\endgroup$ – Daniel Fischer Dec 12 '13 at 19:53
  • $\begingroup$ Oh okay it contains a ball around zero because every point is an interior point. Would I be right in stating that not all subspaces of Hilbert spaces are closed? I read some texts which stated that Hilbert spaces are complete so the subspaces are required to be complete and therefore closed? $\endgroup$ – user103184 Dec 12 '13 at 19:57
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    $\begingroup$ Yes, every infinite-dimensional Hilbert space (that holds for more general classes of topological vector spaces) contains subspaces that aren't closed. Every finite-dimensional subspace of a (Hausdorff) topological vector space is closed. A subspace of a complete space is closed if and only if it is complete. $\endgroup$ – Daniel Fischer Dec 12 '13 at 20:02
  • $\begingroup$ Thanks, one last question, when referring to subspaces of normed vector spaces is the convention always to assume it is a linear subspace with induced topology from the norm and not just any subset with induced norm topology? $\endgroup$ – user103184 Dec 12 '13 at 20:07
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    $\begingroup$ Unless the author explicitly stated that by subspace (s)he refers to a subspace in the sense of topology (a subset with the induced topology), it is safe to assume that custom is followed and subspace means linear subspace (with the induced topology) in this context. $\endgroup$ – Daniel Fischer Dec 12 '13 at 20:09
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Question was answered in comments by Daniel Fischer:

  1. A subset can be open without containing a ball around $0$. A subspace, in the sense of linear subspace, always contains $0$, and if it is open, it therefore contains a ball around $0$. And therefore it is then the entire space.

  2. Every infinite-dimensional Hilbert space (that holds for more general classes of topological vector spaces) contains subspaces that aren't closed. Every finite-dimensional subspace of a (Hausdorff) topological vector space is closed. A subspace of a complete space is closed if and only if it is complete.

  3. Unless the author explicitly stated that by subspace (s)he refers to a subspace in the sense of topology (a subset with the induced topology), it is safe to assume that custom is followed and subspace means linear subspace (with the induced topology) in this context.

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