# Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

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 from Lions and Magenes that $\mathcal{D}([0,T];V) \subset W(0,T)$ 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? I guess I may need to replace $C_c^\infty(0,T)$ by $C_c^\infty[0,T]$..

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

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See also this question: mathoverflow.net/questions/87486/… – András Bátkai Jul 18 '13 at 17:59

We do not know that $C^\infty_c(0,T;V)$ is dense in $W(0,T)$. Note that $W(0,T)$ embeds into $C([0,T],V^*)$ (actually even into $C([0,T],H)$). Since $W(0,T)$ clearly contains functions which do not vanish at the endpoints, no set of functions which are required to vanish at the endpoints can be dense.
Thanks for replying. But is not $C_c^\infty(0,T;V) = \mathcal{D}(0,T;V)$? The latter (defined as infinitely differentiable compactly-supported $V-$valued functions) is dense in $W(0,T)$ by a theorem of Lions and Magenes. – aere Jul 18 '13 at 18:16
Lions and Magenes use the notation ${\cal D}([a,b])$ to denote $C^\infty$ functions with compact support in the closed interval $[a,b]$. The condition of compact support is of course redundant in this case unless the interval is infinite. This should not be confused with ${\cal D}(a,b)$, which is a set of functions which have support which is compact in the open interval $(a,b)$. – Michael Renardy Jul 18 '13 at 19:11