A subspace of $L^p(\mu;E)$ with no nonzero annihilator in $L^q(\mu;E)$ is dense ($\sigma$-finite $\mu$, finite-dimensional $E$, $1<p<\infty$)

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A subspace of $L^p(\mu;E)$ with no nonzero annihilator in $L^q(\mu;E)$ is dense ($\sigma$-finite $\mu$, finite-dimensional $E$, $1<p<\infty$) · Prove2Me