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

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?

• No: it's not true in infinite dimension. Anyway, assuming AC, every subspace of $H$ has a basis. – Crostul Apr 14 '17 at 20:51
• 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. – Hagen Knaf Apr 14 '17 at 20:58
• What sort of basis? A Hamel basis or something like a Hilbert basis? – Daniel Fischer Apr 14 '17 at 20:58
• Possibly useful: math.stackexchange.com/questions/177923/… – Vitor Borges Apr 14 '17 at 21:00
• 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? – paul garrett Apr 14 '17 at 22:17