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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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Sorry, but.. What do you mean by "multi-dimensional" norm? – Berci Oct 11 '12 at 16:27
@Berci I mean the norm of a function with several components. – Ann Oct 11 '12 at 16:28
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? – copper.hat Oct 11 '12 at 16:40
up vote 1 down vote accepted

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 $$ \|(u,v)\|_{L^2} = \|u\|_{L^2} + \|v\|_{L^2} $$

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