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How to prove that $L^p$ space is a subset of $L^q$ space whenever $1 \leq p \leq q < \infty$?

I tried to solve it using the $L^p$ space definition. Tell me how to proceed further?


marked as duplicate by tomasz, user91500, user370967, Davide Giraudo, kingW3 Jun 23 '17 at 11:20

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  • $\begingroup$ your statement is however true for "little" lp and "little" lq spaces (the spaces of convergent series). $\endgroup$ – user135891 Mar 30 '14 at 16:21

This isn't true. Consider $f(x) = \frac 1{\sqrt x}$ which is in $L^1[0,1]$ but not in $L^2[0,1]$ .


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