Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$ Renardy and Rogers says that
1) Functions of the above form are dense in $H^1(0,T;V)$
2) Functions of the above form are dense in $C_c^\infty(0,T;V)$
Why is this true? How to prove it?
Edit: They didn't define $H^1(0,T;V).$ I guess it consists of functions $u(t) \in V$ for each $t$ with $u(t) \in L^2(0,T)$ such that the derivative $u'(t) \in V$ for each $t$ with $u'(t) \in L^2(0,T)$ too.
$C_c^\infty(0,T;V)$ consists of functions $f(\cdot):[0,T] \to V$, so $f(t) \in V$ for each $t$. Furthermore $f$ is $C_c^\infty$ around $[0,T]$ wrt. $t$.