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Consider two Sobolev spaces $H^s(\mathbb R^n)$ and $H^t(\mathbb R^n)$ where $s>t$, if $V$ is a linear subspace of $H^s(\mathbb R^n)$ such that there exists a constant $C$ and any $f \in V$, we have $||f||_{H^s}\leq C||f||_{H^t}$.

Then my question is, is $V$ necessarily finite dimentional?

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Yes, because it would imply that the intersection of the unit ball of $H^t$ with $V$ is compact in $H^t$.