"L^2_loc mod constants" as a reflexive space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:09:41Z http://mathoverflow.net/feeds/question/65215 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65215/l2-loc-mod-constants-as-a-reflexive-space "L^2_loc mod constants" as a reflexive space Bernhard 2011-05-17T09:09:11Z 2011-05-17T09:53:38Z <p>In an article of Sten Kaijser ("A note on dual Banach spaces") I find the assertion that $E = L^2_{\text{loc}}({\mathbb R})$ modulo constants is a reflexive space. </p> <p>Question 1: which is the 'natural' locally convex top. vector space structure of this space? (maybe the proj. limit of $L^2(K_n)$-spaces modulo constants where $(K_n)_{n\ge 1}$ exhausts ${\mathbb R}$ compactly?).</p> <p>Question 2: How can I see reflexivity of $E$? </p> http://mathoverflow.net/questions/65215/l2-loc-mod-constants-as-a-reflexive-space/65218#65218 Answer by Piero D'Ancona for "L^2_loc mod constants" as a reflexive space Piero D'Ancona 2011-05-17T09:53:38Z 2011-05-17T09:53:38Z <p>I'd say the natural structure on $E$ is one of a Frechet space, given by the family of seminorms $\|f\|_{L^2(K_n)}$. Each local space generated by each seminorm is reflexive (Hilbert) so the resulting topology is reflexive. Then if you take the quotient by the closed subspace of constant functions you get again a reflexive space. Details in <a href="http://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fsp.pdf" rel="nofollow">this</a> nice set of lectures by Dietmar Vogt.</p>