most recent 30 from http://mathoverflow.net 2013-06-19T22:39:26Z http://mathoverflow.net/feeds/question/112717 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112717/are-banach-space-ultraproducts-stable-under-infinite-sums Slavoj Žižek 2012-11-17T21:06:27Z 2012-11-17T21:18:22Z

<p>For a pair of Banach spaces $X,Y$ and an ultrafilter $U$ it is easy to find an isomorphism between $(X\oplus Y)_U$ and $X_U\oplus Y_U$. Is this preserved under infinite sums, that is,</p> <p>Let $X$ be an infinite-dimensional Banach space and let $U$ be an ultrafilter over $\mathbb N$. Do we have</p> <p>$$\ell_\infty(X_U) \approx [\ell_\infty(X)]_U?$$</p> <p>$\ell_\infty(X)$ denotes the $\ell_\infty$ sum of countably many copies of $X$.</p>