TITLE: Equivalent Norms on Sobolev Spaces
QUESTION [8 upvotes]: Let $u$ be in the Sobolev space $H^2$. Then the standard definition of the norm is
$$ ||u||_{H^2}^2 = \sum_{|\alpha| \leq  2} ||D^{\alpha} u||_{L^2}^2. $$
However, I have also seen it defined this way
$$ ||u||_{H^2}^2 = ||u||_{L^2}^2 + \sum_{|\alpha| =  2} ||D^{\alpha} u||_{L^2}^2. $$
Are these two norms equivalent? Clearly, the latter is $\leq$ the former. To get an ineqaulity the other way, is the Poincare inequality used?

REPLY [3 votes]: As both are Banach spaces with the respective norm, and the second is $\geq$ than the first, then by the Banach's theorem applied to the identity (The inverse of an invertible
bounded linear operator between Banach spaces is continuous) both are equivalent.