"Orthogonal complement" of a subspace of a Banach space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:00:09Z http://mathoverflow.net/feeds/question/47869 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47869/orthogonal-complement-of-a-subspace-of-a-banach-space "Orthogonal complement" of a subspace of a Banach space Anthony Quas 2010-12-01T05:51:24Z 2010-12-01T14:01:47Z <p>I'm looking for a reference. I am considering the situation where $X$ is a Banach space and $Y$ is a closed finite co-dimensional subspace. I am looking for a $W$ that is a complement of $Y$ (i.e. so that $X$ is the topological direct sum of $W$ and $Y$). I want them to be as far from parallel as possible.</p> <p>The sense that I'm toying with (but I'm happy to have other senses proposed - I suspect they're all fairly equivalent) is that I want to consider the map $\Phi\colon Y\times W\to X$ given by $\Phi(y,w)=y+w$ to have the property that $\|\Phi^{-1}\|$ has small norm (say you put the norm $\|(y,w)\|=\max(\|y\|,\|w\|)$ on the product.</p> <p>So I'm looking for $W$ that makes $\|\Phi^{-1}\|$ small. The question then is: how small can you make this quantity? I think I can make it less than $6^n$ if $Y$ is $n$-codimensional (irrespective of the space $X$), but I presume there are better bounds and references out there. </p> http://mathoverflow.net/questions/47869/orthogonal-complement-of-a-subspace-of-a-banach-space/47880#47880 Answer by Gideon Schechtman for "Orthogonal complement" of a subspace of a Banach space Gideon Schechtman 2010-12-01T08:06:14Z 2010-12-01T08:06:14Z <p>You can make the norm of $\|\Phi^{-1}\|$ to be of order $\sqrt n$. This is basically a theorem of Kadets and Snobar. A good reference is III.B.11 in Wojtaszczyk, P., Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, 25. Cambridge University Press, Cambridge, 1991.</p> http://mathoverflow.net/questions/47869/orthogonal-complement-of-a-subspace-of-a-banach-space/47904#47904 Answer by Juan Valdez for "Orthogonal complement" of a subspace of a Banach space Juan Valdez 2010-12-01T14:01:47Z 2010-12-01T14:01:47Z <p>I think these articles should meet their expectations, at least temporarily</p> <p><a href="http://www.springerlink.com/content/wk3121768n46462x/fulltext.pdf" rel="nofollow">http://www.springerlink.com/content/wk3121768n46462x/fulltext.pdf</a></p> <p><a href="http://arxiv.org/PS_cache/math/pdf/0501/0501048v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/math/pdf/0501/0501048v1.pdf</a></p>