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I've managed to prove that for $ 1\leq p < q \leq +\infty $ we have an inclusion (embedding) $ L_q([0,1],\lambda) \rightarrow L_p([0,1], \lambda) ~~ (\lambda $ being Lebesgue measure). The trouble I'm facing now is:

a) is this embedding continuous? I can imagine why a preimage of an open subset in $ L_p $ should be open in $ L_q $, but I can't really prove that.

b) Is the image of this embedding a dense set? A closed set?

I'm having trouble understanding what open sets in $ L_p $ might be.

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  • $\begingroup$ Unless I'm misunderstanding you, $L_p$ has a norm topology and so - I'm unlikely to make the right choice of terms here between basis and subbasis - is generated by open balls about each point. $\endgroup$ Oct 23, 2014 at 19:21
  • $\begingroup$ Well, of course you're right. The trouble I'm facing is rather finding any properties of the open basis sets such as $ B(f,a)=\{g \in L_p: (\int_{0}^{1}|f-g|^p\})^{1/p} < a $. I simply can't imagine a way to find $ f \in L_p $ such that $ B(f,a)\cap \text{im}(L_q) = \emptyset $ $\endgroup$
    – Jytug
    Oct 23, 2014 at 19:30

1 Answer 1

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We have the relationship $\lVert f\rVert_p\leqslant \lVert f\rVert_q$ for each function $f$. Since the embedding is linear, it follows that it is continuous.

The image contains $\mathbb L^\infty$, which is dense. It can be written as a countable union of closed sets with empty interior.

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  • $\begingroup$ So you say then that $L_q$ is meager in $L_p$? $\endgroup$
    – Arnulf
    May 10, 2016 at 4:11

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