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Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and the space $L^2(0,T;V)$ of square Bochner integrable functions.

It holds that $C_0^\infty(0,T)$ is dense in $L^2(0,T;\mathbb R)$, see, e.g., Cor. 4.23 in Brezi's book on Sobolev Spaces and PDEs .

Does a related result hold in the Bochner space setting, namely, is

$$ C_0^\infty(0,T)\cdot V := \left \{ f_v\in \bigl((0,T)\to V \bigr) :\quad f_v=\phi v, \quad v \in V,\quad \phi \in C_0^\infty(0,T)\right \} $$ dense in $L^2(0,T;V)$?

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  • $\begingroup$ I know it's a bit late but you might want to check the book of Ben Schweizer "Partielle Differentialgleichungen". In Addition there is a nice pde script of John Hunter: math.ucdavis.edu/~hunter/pdes/pde_notes.pdf where also Bochner integrals are introduced $\endgroup$ – Quickbeam2k1 Apr 10 '14 at 10:20
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No. First you should observe that $C_0^\infty(0,T) \cdot V$ is not a subspace. For two (not colinear) vectors $v_1, v_2$, the function $f(t) = \chi_{(0,1/2)}(t) \, v_1 + \chi_{(1/2,1)}(t) \, v_2$ is cannot be approximated.

The set of all finite linear combinations of functions from $C_0^\infty(0,T) \cdot V$ is dense in $L^2(0,T; V)$.

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  • $\begingroup$ Yes, true. Fortunately, the linear hull of $C_0^\infty(0,T)\cdot V$ is still OK for my purposes. Do you know a reference to the latter claim? $\endgroup$ – Jan Oct 7 '13 at 8:17
  • $\begingroup$ Actually, I am trying to prove that the definitions of weak derivatives $\dot v \in L^2(0,T;V')$ of $v\in L^2(0,T;V)$ in the weak sense and in the 'pointwise' sense via $\langle \dot v(\cdot), w \rangle \colon (0,T) \to \mathbb R$, for all $w\in V$, are equivalent. Cf. this question $\endgroup$ – Jan Oct 7 '13 at 8:38
  • $\begingroup$ No, I do not know a reference. However, it should be easy to prove by using an argument involving mollifiers. $\endgroup$ – gerw Oct 7 '13 at 11:14

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