I am studying Michiel's PDE book. Here I have a question about one theorem in chapter about Sobolev space.
At the beginning of this chapter, he define $\tau_y(u)(x)=u(x+y)$ and by assuming $u\in H^s(R^N)$, he conclude without the prove that $$\sigma^{-1}(\tau_{e_j\sigma}u(x)-u(x))\to i\partial_j u \text{ in }H^{s-1}$$ However, I did the calculation myself and I keep getting $$\sigma^{-1}(\tau_{e_j\sigma}u(x)-u(x))\to \partial_j u \text{ in }H^{s-1}$$ i.e., without $i$ in front of $\partial_j u$. I think my result, intuitively, true.(as the sense in classical approach) However, he repeat this result several times so I don't think this is just a typo... So can anybody explain me how this $i$ come from?
Thx!