I'm now reading Pazy's book about the semi-group operator. To prove the existence of the solution of KdV equation. He define the Hilbert space $H^s(\mathbb{R})$ $ \Vert u\Vert_s=\left(\int(1+\xi)^s|\widehat{u}(\xi)|^2d\xi\right)^{\frac{1}{2}}$ the inner product $(u,v)_s=\int(1+\xi)^s\widehat{u}(\xi)\overline{\widehat{v}}(\xi)d\xi)$ where $\widehat{u}$ is the Fourier Transform of $u$. In his book, I have the following question,when $s\geq 3$:
1.If $u\in H^s(R)$,why $Du\in H^{s-1}(\mathbb{R})$? ($Du$ denote $\frac{d}{dx}$)
2.Why $Du\in L^{\infty}(\mathbb{R})$? And $ \Vert Du \Vert_{\infty}\leq C \Vert Du \Vert_{s-1}\leq C \Vert u \Vert_s$?