Is this correct? If $f\in L^1(\mathbb R^d)\cap L^\infty(\mathbb R^d)$ then there exists $(f_n)_{n\in\mathbb N}$ in $C^\infty(\mathbb R^d)\cap L^1(\mathbb R^d)\cap L^\infty(\mathbb R^d)$ such that $f_n\rightarrow f$ in $L^p(\mathbb R^d) $ for every $p\in [1,\infty)$.
I don't how to prove or disprove this. Maybe using the theorem of Riesz-Fischer?
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marc
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You only need to prove that $f_n\to f$ in $L^1$ and $(f_n)$ is bounded in $L^\infty$. – daw Aug 20 '22 at 17:07
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Why do I only have to prove $f_n\rightarrow f$ in $L^1$ and that $f_n$ is bounded in $L^\infty$? @daw – marc Aug 20 '22 at 17:17
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as this implies the claim by Hoelder inequality. – daw Aug 20 '22 at 17:20