# The Equivalents Norms in Sobolev Spaces

Show that over $W^{m,p}(a,b)$ the norms

$$||w||^p_{W^{m,p}(a,b)} = \sum_{i=1}^m\int_a^b|w^{(i)}|^pdx$$

$$||w||^p_{W} = \int_a^b|w|^pdx + \int_a^b|w^{(m)}|^pdx$$

are equivalents.

pdta: By $w^{(j)}$ we denoting the j-ésima derivate of $w$.

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It seems you are commanding the audience... –  paul garrett Aug 2 '13 at 0:19
Do you mean to sum from $1$ or $0$? The function $w(x)=1$ has finite norm in the first and infinite norm in the second. –  robjohn Aug 2 '13 at 0:31