Hilbert space having all norms (and seminorms) continous. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:54:47Z http://mathoverflow.net/feeds/question/56167 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56167/hilbert-space-having-all-norms-and-seminorms-continous Hilbert space having all norms (and seminorms) continous. RadonNikodym 2011-02-21T12:00:19Z 2011-02-21T13:11:12Z <p>Suppose I have a Hilbert space \$H\$ such that every seminorm on \$H\$ is continuous with respect to the inner-product induced norm. Is \$H\$ necessarily finite-dimensional? If not, is there an easy example of an infinite dimensional one?</p> <p>Can anything else be said about such spaces?</p> http://mathoverflow.net/questions/56167/hilbert-space-having-all-norms-and-seminorms-continous/56169#56169 Answer by Sven Raum for Hilbert space having all norms (and seminorms) continous. Sven Raum 2011-02-21T13:11:12Z 2011-02-21T13:11:12Z <p>If every seminorm on \$H\$ is continuous, then every linear functional on H is continuous. Since there are unbounded linear functionals on infinite dimensional Hilbert spaces, \$H\$ must necessarily be finite dimensional.</p> <p>Actually, the locally convex topology on a vector space that makes all linear functionals continuous, doesn't describe anything new. It's probably useful to reformulate algebraic problems (I remember talking to someone about this), but it doesn't give more structure to the vector space.</p>