Equivalent norm in sobolev space H^2

Question

I consider space $H^{2}(0,a)=\{ f\in L^{2}(0,a): f',f''\in L^{2}(0,a) \}$ I define norm $\Vert w \Vert_{H^{2}}:=b\Vert w''\Vert_{L^{2}}$, where b is positive constant.

I couldn't proof that it is norm equivalent to standard norm in $H^{2}$.

Maybe is easier show that $H^{2}$ with this norm is a Hilbert space?

Could you help me?

Answer 1

This is false; consider $w = 1$ (a constant function on the interval). Then clearly $w$ has non-zero $H^2$-norm, but if you just take the $L^2$ norm of its derivative, it'll of course be zero.

However, it is true if you're considering the space $H^2_0(0,a)$, the space of $H^2$ functions with zero trace.