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Let $T : H\rightarrow H$ be a continuous, injective, symmetric operator on a real Hilbert space $H$. Let $V$ be a subspace of $H$ such that $T(V)$ is dense in $H$. Must $V$ be dense in $H$?

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The example you accepted is neither injective nor symmetric. –  Martin Feb 12 '13 at 7:57

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