I have a question about the density of some subspace of $H^1(0,1)$ in it. Let $\alpha$ be a continuous function on $[0,T] \times[0,1]$ where $T>0$ is a fixed positive number.
We consider the subspace of $H^1(0,1)$ $$V(t) = \left\{ {v \in {L^2}(0,1),{\rm{ }}{\partial _x}v + \alpha (t,x)v \in {L^2}(0,1)}, t\in[0,T] \right\}$$ My question is : does $V(t)$ dense in $H^1(0,1)$? Thank you.
The definition of $V(t)$ is flawed. You cannot write $t\in [0,T]$ inside the set because $t$ is already fixed outside. I guess you mean $$ V(t) = \{v\in L^2 : v' + \alpha(t,\cdot)v\in L^2\},\quad t\in [0,T]. $$ But if $v\in L^2$, then also $\alpha(t,\cdot)v\in L^2$ and so the condition $v' + \alpha(t,\cdot)v\in L^2$ is equivalent to $v'\in L^2$. Hence, $V(t) = H^1(0,1)$ for every $t\in [0,T]$.