I met a problem related with proving whether a subspace of Hilbert space is still a Hilbert space.
Given the space $V = \{v \in H^{1}(0,1), \int_0^1 v \mathrm{d} x = 0 \}$, where $v\in H^{1} (0,1)$ iff $v, v' \in L^{2}(0,1)$. So under the norm $ ||v||_{H^{1}}^{2} = \int_{0}^{1} v^{2}(x)+(v^{\prime}(x))^2 \mathrm{d} x$, prove that $V$ is still a Hilbert space.
Any hint will be helpful. Thanks!