# Hilbert space having all norms (and seminorms) continous.

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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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.