Comparable Norm on Banach - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T21:26:16Z http://mathoverflow.net/feeds/question/109028 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109028/comparable-norm-on-banach Comparable Norm on Banach Li 2012-10-06T23:48:53Z 2012-10-07T00:32:08Z <p>Let $\mathcal{X}$ be a vector space, $| \cdot |$ and $\| \cdot \|$ be two norms on which $\mathcal{X}$ is complete with respect to both. Can the two norms be not equivalent? Please just give me some hint, I want to be able to do it myself. Thx!</p> http://mathoverflow.net/questions/109028/comparable-norm-on-banach/109029#109029 Answer by Aryeh Kontorovich for Comparable Norm on Banach Aryeh Kontorovich 2012-10-07T00:06:27Z 2012-10-07T00:06:27Z <p>Take $\mathcal{X}=\ell_1(\mathbb{N})$. Then $\mathcal{X}$ is complete under $\ell_1$ and $\ell_\infty$, while the two norms are not equivalent.</p> http://mathoverflow.net/questions/109028/comparable-norm-on-banach/109032#109032 Answer by Delio Mugnolo for Comparable Norm on Banach Delio Mugnolo 2012-10-07T00:32:08Z 2012-10-07T00:32:08Z <p>Take a look at <a href="http://www.springerlink.com/content/k84641k661604hl2/" rel="nofollow">this article</a>:</p> <p>From the abstract: "[...] on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms".</p>