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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Strong continuity (weak to strong) of $\langle Au,v\rangle=\int u^3 v dx$

I am currently trying to figure out the following. If I consider the Sobolev space $W^{1,p}_0$ is it possible to show that the operator given by $$\langle Au,v\rangle=\int u^3 v dx$$ is strongly (weak …
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