I need help with this problem from my homework
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's constants $M, \beta > 0$, such that $\beta \leq \kappa(x) \leq M$ for each $x\in\Omega$. Now, consider $$\langle u, v\rangle_{\kappa}\ :=\ \int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v,\;\;\; \forall\ u,v\in H^1(\Omega),$$ and show that
- $\langle \cdot, \cdot\rangle_{\kappa}$ is an inner product of $H^1(\Omega) = \{v\in L^2(\Omega) : \nabla v\in L^2(\Omega)\}$,
- $\|\cdot\|_{\kappa} = \sqrt{\langle \cdot, \cdot\rangle_{\kappa}}$, is equivalent with the norm $$\|u\|_{H^1(\Omega)}\ = \ \left(\int_{\Omega}u^2\ + \int_{\Omega}\nabla u\cdot\nabla u\right)^{1/2}.$$
Please somebody can help me. Thanks in advance.