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Can you help me, plese, with the notion of closed linear subspace. What means, examples of closed linear subspace, how can I prove that a subspace is a closed linear subspace.

Thanks :-)

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    $\begingroup$ Where? In a normed vector space? $\endgroup$
    – Julien
    Commented Jan 13, 2013 at 15:21
  • $\begingroup$ @julien Hilbert spaces! $\endgroup$
    – Iuli
    Commented Jan 13, 2013 at 15:24

1 Answer 1

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A linear subspace $V$ of a Hilbert space $H$ is called closed it is closed with respect to the norm topology: i.e. whenever $(v_n)$ in $V$ converges to $h$ ($\| v_n-h\|\rightarrow 0$) in $H$, then $h$ belongs to $V$.

Examples: any finite-dimensional subspace is closed, any orthogonal $A^\perp$ is a closed linear subspace.

Counter-example: if $(e_n)$ is a Hilbert basis of an infinite-dimensional separable Hilbert space $H$, then $V=Vect((e_n))$ is a linear subspace of $H$ which fails to be closed.

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