= Clamped interval derivative norm bound
{title2=$\|v\|_{H^2(-1,1)}^2\leq21\|v''\|_2^2$}
For $v\in H_0^2(-1,1)$, integrate its <weak derivative> from the left endpoint, where $v=v'=0$. The <Cauchy-Schwarz inequality> gives $\|v'\|_2\leq2\|v''\|_2$ and $\|v\|_2\leq2\|v'\|_2$. Adding the three squared derivative norms gives the displayed bound. The constant is sufficient, rather than asserted sharp. Thus the second-derivative norm is equivalent to the full <Sobolev norm> on this clamped space.
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