Let $V \subset H \subset V^*$ be separable Hilbert spaces with continuous and dense embeddings. Define the Hilbert space $$W(0,T) = \{u \in L^2(0,T;V) : u' \in L^2(0,T;V^*)\}$$ with inner product $$(u,v)_W = \int_0^T (u(t),v(t))_{L^2(0,T;V)} + \int_0^T (u'(t), v'(t))_{L^2(0,T;V^*)}.$$

I want to know whether the set of functions of the form $$w(t) = \sum_j \phi_j w_j, \qquad\text{where $\phi_j \in C_c^\infty(0,T)$ and $w_j \in V$}$$ are dense in $W(0,T).$

We know that $C_c^\infty(0,T;V) \subset W(0,T)$ is dense, so the above should hopefully be true. According to a book, the set of functions $$f(t) = \sum_j t^j w_j \quad \text{where $w_j \in V$}$$ are indeed dense in $W(0,T)$.

Does this imply the result I want? Can I approximate the $t^j$ by $C_c^\infty(0,T)$ functions or something like that? (I don't think so). Or is there another way to do this? Thanks.

I posted this at math.stackexchange.com but got no answers.