0
$\begingroup$

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.

  • 0
    Sorry, but.. What do you mean by "*multi-dimensional*" norm?2012-10-11
  • 0
    @Berci I mean the norm of a function with several components.2012-10-11
  • 0
    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

1 Answers 1

1

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