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Let $s\in\mathbb{R}$. The homogeneous Sobolev space $\dot{H}^{s}(\mathbb{R}^n)=\left\{u\in \mathcal{S}': \left\|u\right\|^2_{s}=\int_{\mathbb{R}^n}|\xi|^{2s}|\widehat{u}(\xi)|^2\,d\xi<\infty\right\}$

Question If $s=2$. Why $\left\|u\right\|_{2}$ is a norm if when $\left\|u\right\|_{2}=0$ then $|\xi|^{2}\widehat{u}=0$ but this implies that $u$ is a polynomial in $(x_1,\ldots, x_n)$ (Taylor. PDE Basic theory). What am I misunderstanding?

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