# $\mathcal{D}(0,T;V)$ is dense in $W(0,T)$

Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(0,T;V^*)\},$$ where $V$ is Hilbert\Banach? I need the proof in the English language.

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What is $T$? What do the $0$s mean? What is $\mathcal{D}$? Is this question research-level? –  Benoît Kloeckner May 10 '13 at 16:55
$[0,T]$ is an interval. This is a research-level question. –  maximumtag May 10 '13 at 17:11
A standard reference on this is the monograph by Lions and Magenes: Non-Homogeneous Boundary Value Problems and Applications. Everyone refers to it for the proof... –  András Bátkai May 10 '13 at 17:55
Volume 1: rd.springer.com/book/10.1007/978-3-642-65161-8/page/1 , but there are three. –  András Bátkai May 10 '13 at 17:57
@AndrasBatkai thank you a lot. Unfortunately I don't have access to that. Do you have a PDF of that (if legal, otherwise obviously ignore this)? –  maximumtag May 10 '13 at 21:04