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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.

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    It might be easier if you give a specific example of what you are interested in. Are you asking how the specific norms you mentioned are defined?2012-10-11

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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} $