Strong topology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:55:31Z http://mathoverflow.net/feeds/question/72106 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72106/strong-topology Strong topology Celeban 2011-08-04T18:09:05Z 2011-08-16T21:27:26Z <p>Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively. A dual of $E^{\star}$ given by the $\beta(E^{\star\star}, E^{\star})$ topology is usually denoted by $E^{\star\star}$ and it is called a double dual of $E$.</p> <p><strong>Question 1</strong></p> <p>Is it true that the topology $\beta(E^{\star}, E^{\star\star})$ on $E^{\star}$ is always finer than $\beta(E^{\star}, E)$ (on $E^{\star}$), apart from the case when $E$ is reflexive and these topologies coincide. </p> <p><strong>Fact 1</strong></p> <p>It is well-known that if $E$ is an (F)-space then the (initial) (F)-space topology coincides with strong topology $\beta(E, E^{\star})$.</p> <p><strong>Question 2</strong></p> <p>Can we find a non-metrizable locally convex space for which the above sentence is true? (i.e. instead of (F)-space we put a non-metrizable locally convex space).</p> <p>Thank you in advance for any help.</p> http://mathoverflow.net/questions/72106/strong-topology/72136#72136 Answer by Đức Anh for Strong topology Đức Anh 2011-08-05T02:03:15Z 2011-08-05T02:22:42Z <p>For the first question, the strong topology is the polar topology generated by all weakly bounded subsets. The weakly bounded subsets of $E$ are also weakly bounded in $E^{\ast\ast}$ since $E\subset E^{\ast\ast}$ and they have the same dual space $E^{\ast}.$ Therefore $\beta(E^{\ast},E^{\ast\ast})$ is finer than $\beta(E^{\ast},E).$</p> <p>For the second question, did you try the space of test functions, i.e, infinitely differentiable functions with compact support? More generally, Hausdorff barrelled spaces have the property you want, so you should look for spaces in the class of barrelled spaces.</p>