I am working in Navier-Stokes Equations by Roger Temam and he uses some notions that are not very clear to me:
For $V$ a Hilbert space and $V'$ its dual, we are considering $u \in L^2(0,T;V)$, where $$L^2(0,T;V) = \left\{u:[0,T] \to V~\bigg|~ \int_0^T \|u(t)\|^2_V~dt< + \infty\right\},$$ and the author says that "its derivative $u'$ belongs to $L^2(0,T;V')$". But what does he exactly mean by "its derivative" $?$ I am familiar with the concept of weak derivative of function in $L^2(\mathbb R)$ but as here our functions are valued in $V$, I am not totally comfortable with this notion. Does that mean that there is some function $u'\in L^2(0,T;V')$ such that, for all $ \varphi\in C^\infty_0([0,T])$ (compactly supported functions valued in $\mathbb R$) $$\int_0^T u \partial_t\varphi~dt = - \int_0^Tu' \varphi~dt,$$ or that for all $ \varphi\in C^\infty_0([0,T], V)$ (compactly supported functions valued in $V$) $$\int_0^T (u, \partial_t\varphi)_V~dt = - \int_0^T(u', \varphi)_V~dt ~?$$ Where $(\cdot, \cdot)_V$ the scalar product in $V$.
