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I am wondering if there exist examples of Banach (or Frechet) sequence spaces in which the set of all finite support sequences are NOT dense?

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migrated from Jun 18 '14 at 20:12

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whatever you mean by "finite support sequences" in a Banach space X, you can consider the direct sum of X and the line generated by a hen. – Pietro Majer Jun 18 '14 at 15:07
@Pietro: I thought hens generated ellipsoids, not lines ... – Nik Weaver Jun 18 '14 at 18:59

Probably the simplest example is $c$, the space of convergent sequences. The closure of the finitely supported sequences is, of course, $c_0$, the space of null sequences.

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