I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
References on relationships between different $L^p$ spaces
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real-analysis
general-topology
reference-request
measure-theory
banach-spaces
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0@commenter Thanks! The title is indeed inncocent – 2012-10-18
3 Answers
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I learned about that stuff from Kolmogorov and Fomin's Real Analysis text, I believe. It has the benefit that it's cheap for a math book.
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0Thanks Shawn. I was not really looking for a real analysis textbook but for a list of known results and counter-examples, and where to find them in particular. That would be great if that particular textbook contains all of those. Does it? – 2012-10-18
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Here is a special case, we will prove $ L_2[a,b] \subset L_1[a,b] \,.$ Assume $f \in L_2[a,b]\,,$ then
$ \int_{a}^{b}|f(x)|\,dx \leq\sqrt{\int_{a}^{b}1\,dx} \sqrt{\int_{a}^{b}f(x)^2\,dx} =\sqrt{b-a}||f||_2 < \infty\,. $
The above inequality follows from the Schwartz inequality or more generally (Holder's inequality).
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2Why don't you post proof for more general case? – 2012-10-18
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I think that my answer here, borrowed from an exercise in Folland's book, addresses this; it gives necessary and sufficient conditions for each inclusion to hold.