Let $$V = \left\{ {v \in {H^2}(0,1):{v_x}\left( 0 \right) - v\left( 0 \right) = {v_x}\left( 1 \right) + v\left( 1 \right) = 0} \right\}.$$

Question: Is $V$ dense in ${H^1}\left( {0,1} \right)$?

I know ${C^\infty }\left( \mathbb{R} \right)$ is dense in ${H^1}\left( {0,1} \right)$. But I can't prove that $V$ is dense in ${H^1}\left( {0,1} \right)$ .