Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^\*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^\* = H^{-1}(\mathbb{R}^n)$.) We can then define the vector-valued Sobolev space $W^{1,2}([0,1]; V, V^\*)$ of functions $u \in L^2([0,1]; V)$ which have one weak derivative $u' \in L^2([0,1], V^\*)$. Such spaces arise often in the study of PDE involving time.


I would like a reference for some simple facts about $W^{1,2}$. For example:


* Basic calculus, like integration by parts, etc.
* The "Sobolev embedding" result $W^{1,2} \subset C([0,1]; H)$;
* The "product rule" $\frac{d}{dt} \|u(t)\|\_{H^2} = (u'(t), u(t))\_{V^\*, V}$
* $C^\infty([0,1]; V)$ is dense in $W^{1,2}$.


These are pretty easy to prove, but they should be standard and I don't want to waste space in a paper with proofs.


Some of these results, in the special case where $V$ is Sobolev space, are in L. C. Evans, *Partial Differential Equations*, section 5.9, but I'd rather not cite special cases. Also, in current editions of this book, there's a small but significant gap in one of the proofs (it is addressed briefly in the latest errata). So I'd prefer something else.


Thanks!



Reference request: Simple facts about vector-valued Sobolev space
J. Wloka "Partial differential equations", § 25 (p. 390 on, in my 1992 CUP edition) has an account of the space $W(0,T)=W\_2^1(0,T)$ which is essentially the space $W^{1,2}([0,T];V,V^\*)$.


