How can we define multi-dimensional norms? For example,
$$ \| (v_1, v_2, \cdots , v_n) \|_{W^{1,2}(X)} \;\;\text{or} \;\;\|(v_1 , v_2 , \cdots , v_n ) \|_{L^2 (X)}$$ for some appropriate functions $v_i$'s.
How can we define multi-dimensional norms? For example,
$$ \| (v_1, v_2, \cdots , v_n) \|_{W^{1,2}(X)} \;\;\text{or} \;\;\|(v_1 , v_2 , \cdots , v_n ) \|_{L^2 (X)}$$ for some appropriate functions $v_i$'s.
Usually, you would use some standard norms like in $\mathbb{R}^n$. For instance, $$ \|(u,v)\|_{L^2} = \sqrt{\|u\|_{L^2}^2 + \|v\|_{L^2}^2} $$ or $$ \|(u,v)\|_{L^2} = \|u\|_{L^2} + \|v\|_{L^2} $$