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Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some $p_0>1.$

I guess the problem is that in infinite measure spaces the inclusion $L^p\subset L^q$ only holds for $p=q$. Is it maybe possbile to apply Baire's Theorem in some way?

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    To go further: does anyone have an example where $X$ is infinite-dimensional?2012-07-07
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    What do you think about defining $Y:=\{\lvert f \lvert\ \ \ \lvert \ \ f \in X\}\cup X$, then using your first proof to conclude that $Y\subset L^q$ and therefor also $X$. The only problem might be, to prove that $Y$ is also closed.2012-07-10

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