I am currently reading works about numerical method for solving differential equations. The main setting of the work revolves on the space $H^m_0[0,1]$, which is defined by $$ H^m_0[0,1]:=\{f\in W^{2,m}[0,1]\ |\ f^{(j)}(0)=f^{(j)}(1)=0\ ;\ j=0,1,\dots,m-1\}\ . $$ It is claimed that $\|u\|_m:=\sqrt{\langle u,u \rangle_m}$ is indeed a norm on $H^m_0[0,1]$, where $\langle u,v \rangle_m$ is defined by $$ \langle u,v \rangle_m:=\int^1_0u^{(m)}(t)v^{(m)}(t)dt $$ (our field is Real). I want to know the relation between this norm and the usual norm induced by $$ \langle u,v \rangle:=\sum_{j=0}^m\left(\int^1_0u^{(j)}(t)v^{(j)}(t)dt\right)\ . $$ Where can I read more about the Sobolev space with homogeneous boundary condition (I don't know if it's normally called by this name)? Is there an introductory article about this and related topics?
Any help would be highly appreciated, thank you in advance.