Banach spaces with simple best approximate solutions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:00:40Z http://mathoverflow.net/feeds/question/85921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85921/banach-spaces-with-simple-best-approximate-solutions Banach spaces with simple best approximate solutions Ricky Demer 2012-01-17T19:25:52Z 2012-01-17T19:25:52Z <p>Let $\langle V,||.||\rangle$ be a Banach space such that: <br><br><br> $\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$ <br> $\;\;$ that minimizes $\langle ||L(u)+(-v)||,||u||\rangle$ in the lexicographic order <br><br> $\;\;$ and <br><br> $\;\;\;$ for all continuous linear maps $\: L : V\to V \:$, $\:$ the function $\: L^{\dagger} : V\to V \:$ given by <br> $\;\;$ $\big[L^{\dagger}(v)$ minimizes $\langle ||L(L^{\dagger}(v))+(-v)||,||L^{\dagger}(v)||\rangle$ in the lexicographic order$\big]$ <br> $\;\;\;$ is continuous and linear <br><br><br><br> Does it follow that $\langle V,||.||\rangle$ is <br><br> $\;\;$ 1. $\:$ isometrically <br> $\;\;$ 2. $\:$ homeomorphically <br><br> isomorphic to a Hilbert space? <br><br></p>