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I am studying the dual space of Lp on Rudin. The theorem in question is 6.16. on page 127, at some point this statement is made:

They coincide on the dense subset $L^\infty(\mu)$ of $L^p(\mu) [\dots]$

In which part of the text is it shown that $L^\infty(\mu)$ is dense in $L^p(\mu)$? Could it be that this statement was proved directly in the proof of the theorem? Could someone who has read this time but about this book help me? Thanks!

NatMath
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  • Did you try to search the site for an answer? https://math.stackexchange.com/questions/662590/why-is-l1-cap-l-infty-dense-is-in-lp is one example. It is the first hit if you google "l infinity dense in lp". – Pedro Jun 10 '22 at 11:33
  • @Pedro Yes, I had already found it. My question is another though: find this answer in Rudin's book. – NatMath Jun 10 '22 at 11:52
  • Rudin expects readers to supply such proofs on their own. – geetha290krm Jun 10 '22 at 12:12

2 Answers2

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This is very elementary and Rudin does not state it as a Theorem. If $f \in L^{p}$ then $\int |f\chi_{|f|\leq n} -f|^{p}=\int |f|^{p}\chi_{|f|> n} \to 0$ by DCT and $f\chi_{|f|\leq n}$ is an $L^{\infty}$ function for each $n$.

geetha290krm
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Simple functions that vanish off a set of finite measure are dense in $L^p$ (follows from the pointwise approximation by simple functions). Such functions are in $L^{\infty}$.

Mason
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