User simen k. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:23:05Z http://mathoverflow.net/feeds/user/25950 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118611/norms-agreeing-on-dense-subspace Norms agreeing on dense subspace Simen K. 2013-01-11T09:30:43Z 2013-01-12T05:37:52Z <p>Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$.</p> <p>Is $(B,\|\cdot\|)$ a completion of $(V,\|\cdot\|_V)$ with respect to the $\|\cdot\|_V$ topology? I.e., can the spaces be considered the same/identical up to some isometry?</p> http://mathoverflow.net/questions/118611/norms-agreeing-on-dense-subspace/118614#118614 Comment by Simen K. Simen K. 2013-01-11T10:40:49Z 2013-01-11T10:40:49Z Great answer, thanks! http://mathoverflow.net/questions/118611/norms-agreeing-on-dense-subspace Comment by Simen K. Simen K. 2013-01-11T10:32:19Z 2013-01-11T10:32:19Z Thanks, yes, I meant a completion by Cauchy sequences, or at least that the completion and $B$ can be considered the same space via isometry.