Suppose that $H$ is an infinite dimensional normed vector space, is every infinite dimensional subspace of $H$ with a basis closed?

In finite dimension, this is correct, but in infinite dimension, is this right?

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    $\begingroup$ No: it's not true in infinite dimension. Anyway, assuming AC, every subspace of $H$ has a basis. $\endgroup$ – Crostul Apr 14 '17 at 20:51
  • $\begingroup$ Some additional facts that may help to understand the situation in infinite dimenson: a proper subspace is never open. The closure of every subspace is itself a subspace. $\endgroup$ – Hagen Knaf Apr 14 '17 at 20:58
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    $\begingroup$ What sort of basis? A Hamel basis or something like a Hilbert basis? $\endgroup$ – Daniel Fischer Apr 14 '17 at 20:58
  • $\begingroup$ Possibly useful: math.stackexchange.com/questions/177923/… $\endgroup$ – Vitor Borges Apr 14 '17 at 21:00
  • $\begingroup$ Seconding @DanielFischer's request for clarification: what sort of basis? An algebraic basis? (Which always exists, by Axiom of Choice), or some presumed more-general analogue of an orthonormal "Hilbert space" basis in a Hilbert space? (Plus, not assuming your space is complete complicates the question even for pre-Hilbert spaces...) Can you clarify? $\endgroup$ – paul garrett Apr 14 '17 at 22:17

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