= Sobolev duality
{c}
{title2=$(H^s(\mathbb R^d))^*=H^{-s}(\mathbb R^d)$}
The continuous <dual space> of $H^s(\mathbb R^d)$ is $H^{-s}(\mathbb R^d)$ under the extension of the <L2 inner product>. The <Fourier transform> definition $\|f\|_{H^s}^2=\int(1+|\xi|^2)^s|\widehat f(\xi)|^2d\xi$ proves boundedness of the pairing by the <Cauchy-Schwarz inequality>, and the <Riesz representation theorem> proves that every continuous functional has this form. Consequently a bounded <linear operator> $K:H^r\to H^{r+a}$ has an <adjoint operator>, relative to this pairing, from $H^{-r-a}$ to $H^{-r}$. For an operator that smooths by two orders for every $r$, this yields $K^*:H^2\to H^4$ by taking $r=-4$.
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