A separable Banach space and a non-separable Banach space having the same dual space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:14:08Z http://mathoverflow.net/feeds/question/77383 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77383/a-separable-banach-space-and-a-non-separable-banach-space-having-the-same-dual-sp A separable Banach space and a non-separable Banach space having the same dual space? Valerio Capraro 2011-10-06T18:40:43Z 2011-10-25T12:34:33Z <p>I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is just that nobody thought enough about that, or maybe it is not a stupid question.</p> <p><strong>Question:</strong> Do there exist two Banach spaces, one separable and one non-separable, having isomorphic dual spaces?</p> <p>Note: isomorphic in the sense that there exists a linear homeomorphism between the two.</p> http://mathoverflow.net/questions/77383/a-separable-banach-space-and-a-non-separable-banach-space-having-the-same-dual-sp/77385#77385 Answer by Bill Johnson for A separable Banach space and a non-separable Banach space having the same dual space? Bill Johnson 2011-10-06T18:56:46Z 2011-10-06T18:56:46Z <p>The duals of $C[0,1]$ and of <code>$C[0,1]\oplus_\infty c_0(\Bbb{R})$</code> are isometrically isomorphic.</p> http://mathoverflow.net/questions/77383/a-separable-banach-space-and-a-non-separable-banach-space-having-the-same-dual-sp/77386#77386 Answer by Philip Brooker for A separable Banach space and a non-separable Banach space having the same dual space? Philip Brooker 2011-10-06T18:59:43Z 2011-10-06T18:59:43Z <p>The James Tree space $JT$ and $JT \oplus_2 \ell_2(2^{\aleph_0})$ have isomorphic duals.</p> http://mathoverflow.net/questions/77383/a-separable-banach-space-and-a-non-separable-banach-space-having-the-same-dual-sp/79069#79069 Answer by Qingping Zeng for A separable Banach space and a non-separable Banach space having the same dual space? Qingping Zeng 2011-10-25T12:29:11Z 2011-10-25T12:34:33Z <p>To the best of my knowledge, among classical Banach spaces, $c_0,$ C[a,b], $L_1[a,b],$ $l_{\infty}/c_0$ are not dual. </p>