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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?

Can anything else be said about such spaces?

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up vote 8 down vote accepted

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.

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.

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"there are unbounded linear functionals on infinite dimensional Hilbert spaces": doesn't this need the Axiom of Choice, or some other slightly weaker logical axiom? So if you don't believe AC then it may be possible to give infinite-dimensional examples. Of course, most people don't dispute the validity of AC very much, so this is probably not an issue for most readers. – Zen Harper Feb 22 '11 at 9:12

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