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Closure and span

Suppose that X is a normed vector space and S is a subset of X. Prove that \overline{span(S)}=\overline{span(\overline{S})}.

Proof We have that S\subset span(S). Then obviously \overline{S}\subset \overline{span(S)}. Since the closure of a subspace is also a subspace we have also span(\overline{S})\subset \overline{span(S)}. Taking the closure in this last inclusion we get
\overline{span(\overline{S})}\subset \overline{span(S)}.
On the other hand, we obviously have S \subset \overline{S}, which implies span(S)\subset span(\overline{S}), which implies \overline{span(S)}\subset \overline{span(\overline{S})}. This proves the desired equality.

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  1. January 8, 2011 at 12:09 am

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