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I'm reading a one proof. This say

If $u$ is a test function (smooth function with compact support), then $$|\delta_0(u)|=|u(0)|=\left|\int_{—1}^0u'(t)dt\right|\leq \lVert u'\rVert_p\leq \lVert u\rVert_{W^{1,p}(-1,1)}.$$ By density of test function, we can extend $\delta_0$ to the functions of $W^{1,p}(-1,1)$.

I don't understand this "By density of test function, we can extend $\delta_0$ to the functions of $W^{1,p}(-1,1)$".

Please, Anybody will be able to explain me?

  • 1
    It means that the test function space $C^{\infty}_c(I)$ is dense in $W^{1,p}_0(I)$ which is a closed subspace of $W^{1,p}(I)$.2012-11-08
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    but Why "we can extend $\delta_0$ to the functions of $W^{-1,p}(-1,1)$"?2012-11-08
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    $W^{-1,p}$ is the space of bounded linear functional on $W^{1,p}_0$, aka the dual space of $W^{1,p}_0$, please see my answer.2012-11-08

1 Answers 1