closed subspace of $W^{1,2}(0,1)$

Question

let $V_{C} = \{u \in W^{1,2}(0,1)\,\, | \,\, u(1) = Cu(0)\}, \,\, C \in \mathbb{R}$

1. show that $V_C$ is a closed subspace of $W^{1,2}(0,1)$

I got stuck trying to solve this here's my work : if we let $T : W^{1,2}(0,1) \to \mathbb{R}$ such that $T(u) = u(1)-Cu(0)$

then $V_C = \ker T$ and $T(u) = C(u(1) - u(0)) + (1-C)u(1) \implies |T(u)| \leq ||u'||_{L^1(0,1)} + K||u||_{\infty} $

we're not in a finite dimension space so I can't move forward

any help will be appreciated.