I know that the Sobolev space $H^s$ with the inner product given by $ \langle u,v \rangle_{H^s} := \sum_{| \alpha| \leqslant s} \int_{\Bbb R^n} \nabla^\alpha u \cdot \nabla^\alpha v $ is Hilbert space. ($u = (u_1, \cdots , u_N), v = (v_1, \cdots ,v_N)$)
Then if we set the bracket as $ \langle u,v \rangle_{A,H^{s}} := \sum_{| \alpha| \leqslant s} \int_{\Bbb R^n} (A \nabla^\alpha u) \cdot \nabla^\alpha v $
with $N \times N$ positive definite matrix $A$, then is $(H^s, \langle \cdot , \cdot \rangle_{A, H^{s}} ) $ also Hilbert space?