are the smooth vectors of a Frechet space dense? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:22:35Z http://mathoverflow.net/feeds/question/91104 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91104/are-the-smooth-vectors-of-a-frechet-space-dense are the smooth vectors of a Frechet space dense? Yul Otani 2012-03-13T19:25:29Z 2012-03-14T08:10:56Z <p>Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms ${ \| \cdot \|_j }$, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is non-empty? Is there any requirements on $G$ or $\alpha$ or $B$? For the case I am (supposed to be) working on right now, we also have that $\alpha$ is isometric for all the seminorms and strongly continuous. </p> <p>Also, since I am almost illiterate on the subject of Lie groups, I would also be thankful for some easy to read references! Thank you so much.</p> http://mathoverflow.net/questions/91104/are-the-smooth-vectors-of-a-frechet-space-dense/91109#91109 Answer by Johannes Ebert for are the smooth vectors of a Frechet space dense? Johannes Ebert 2012-03-13T20:10:43Z 2012-03-13T20:10:43Z <p>If the group $G$ is compact, then your statement is true. More precise, let $A$ be a locally convex complete topological vector space and $G \times A \to A$ be a continuous action of $G$. Let $A_s \subset A$ be the sub vector space generated by finite-dimensional $G$-stable subspaces. Then $A_s$ is dense in $A$. This is a generalization of the Peter-Weyl theorem, compare Bröcker, tom Dieck, Representation of compact Lie groups, Theorem 5.7, p.141. </p> <p>Vectors in $A_s$ are smooth, because actions of Lie groups on finite-dimensional spaces are smooth. So your question follows from that. </p> http://mathoverflow.net/questions/91104/are-the-smooth-vectors-of-a-frechet-space-dense/91145#91145 Answer by paul garrett for are the smooth vectors of a Frechet space dense? paul garrett 2012-03-14T02:05:50Z 2012-03-14T02:05:50Z <p>The answer is "yes" quite generally, and the following argument can be found in many texts (Knapp, Wallach, Varadarajan, ...): for a test function $f$ on the Lie group $G$, there is the "averaged" action on any quasi-complete, locally convex repn space $\pi,V$ for $G$, namely, $f\cdot v=\int_G f(g)\,\pi(g)(v)\;dg$. These images are smooth vectors. The collection of all such is the Garding subspace. Taking $f$'s to run through an _approximate_identity_ shows density.</p> <p>(In fact, Dixmier and Malliavin showed that the Garding subspace is the whole space of smooth vectors.)</p> http://mathoverflow.net/questions/91104/are-the-smooth-vectors-of-a-frechet-space-dense/91152#91152 Answer by Laurent Berger for are the smooth vectors of a Frechet space dense? Laurent Berger 2012-03-14T08:10:56Z 2012-03-14T08:10:56Z <p>In full generality, the answer is "no". If you have a unitary irreducible representation of a $p$-adic Lie group on a $p$-adic Banach space, then the locally constant vectors are usually not dense. There may even be no nonzero locally algebraic vector. The locally constant vectors are dense however if you have an $\ell$-adic Lie group acting on a $p$-adic space, with $\ell \neq p$ (this is a theorem of Vigneras). </p>