What is the usual norm of $\mathbb{H_0^1}$?

Question

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$?

In my work I need to prove that the norm endowed by the inner product $$ ((u,v))=\int_{\Omega} \displaystyle \sum_{j=1}^{2}\nabla u_j \cdot \nabla v_jdx, \ \ u=(u_1,u_2),, \ v=(v_1, v_2) \in \mathbb{H_0^1} $$

it is equivalent to the usual norm. Help, please.

Answer 1

There is a general way to endow the (set theoretic) product of two Hilbert spaces with a norm that makes it a Hilbert space - $||\cdot ||_{H \times K}=\sqrt{||\cdot ||_H^2+||\cdot ||_K^2}$. You can check that this does in fact make $H \times K$ a Hilbert space. You could also check that the topology given by this norm coincides with that given by the topological product of the spaces $H,K$.