relation between norm of spaces

Question

If we have the following space $V^{1}(0,l)$:={$u\in H^{1}(0,l);u(0)=0$}

Is there any relation between the norm of the above space and the norm over the space $H^{1}(0,l)$(an equivalence between the norms for example!)

Answer 1

$V^1(0,l)$ is a subspace of $H^1(0,l)$, so they of course have the same norm... in other words, $\|v\|_{V^1}=\|v\|_{H^1}$ for all $v\in V^1(0,l)$.