Let $\Omega\subset \mathbb{R}^d$ be an open and bounded set. Let $\partial \Omega=\Gamma_{D}\cup\Gamma_{N}\cup \Gamma_{C}$ (disjoint). Consider
$V=\{v\in H^{1}(\Omega, \mathbb{R}^d)\, | v=0 \, \text{on}\, \Gamma_{D}\}$.
Why is $V$ a closed subspace of $H^{1}(\Omega, \mathbb{R}^d)$?