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I've found two answers which seem to contradict eachother;

This first one makes me 100% sure that its not the case, Closure of the span in a Banach space

But I cant figue out why the second one almost also do..when it shouldt

https://mathoverflow.net/questions/104672/infinite-linear-span-vs-closed-linear-span

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user123124
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  • The second one shows: the closed linear span of a linear subspace of a normed space is the set of sums of infinite series. The first one shows it fails when the subset is not already a linear space. – GEdgar Oct 11 '15 at 19:06
  • @GEdgar do you mind clarify "the set of sums of infinite series" ? – user123124 Oct 12 '15 at 19:51
  • If you do not understand one of those links, why not ask there? – GEdgar Oct 12 '15 at 22:40
  • @GEdgar I think I understand it, but "set of sums of infinite series" made me think twice. According to me the second is using "series of linear combainations from A" when constructing the sequence. Sums of series would be a finite number of infinte sums added together which I dont think he is doing. – user123124 Oct 13 '15 at 06:33
  • By "sum of an infinite series" I mean "limit of the partial sums of the series" – GEdgar Oct 13 '15 at 12:25

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As GEdgar points out ;

The second one shows: the closed linear span of a linear subspace of a normed space is the set of sums of infinite series. The first one shows it fails when the subset is not already a linear space. By "sum of an infinite series" he means "limit of the partial sums of the series"

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