Let $$V = \left\{ {v \in {H^2}:{v_x}\left( 0 \right) - v\left( 0 \right) = {v_x}\left( 1 \right) + v\left( 1 \right) = 0} \right\}.$$ Prove or disprove that $V$ is dense in ${H^1}\left( {0,1} \right)$. I know ${C^\infty }\left( \mathbb{R} \right)$ dense in ${H^1}\left( {0,1} \right)$. But i cant prove that $V$ is dense in ${H^1}\left( {0,1} \right)$ .