Why is density and separability needed for uniqueness of weak (time) derivatives? Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\infty(0,T).$$
Suppose that $u$ also has a weak derivative $h \in L^1(0,T;Z)$ where $Y \subset Z$.
In Boyer and Fabrie's book on Navier-Stokes, page 95, he states that if $Y \subset Z$ is dense and $Z'$ is separable, then $g=h$. My question, why is the density and separability needed? Isn't the argument this simple:
Since $g$ and $h$ are weak derivatives of $u$, we have
$$\int_0^T (g(t)-h(t))\phi(t) = 0\qquad\forall \phi \in C_c^\infty(0,T)$$
and by the fundamental lemma of the calculus of variations, it follows that $g(t) = h(t)$ in $Z$ for almost every $t$.
Isn't this enough? What do I miss? Does anyone know another source for this uniqueness claim where the derivatives lie in different spaces?
 A: $\def\bbR{\mathbb R}\def\inc{\subseteq}$The requirements on density or separability are superfluous because of the following
Lemma. Let $J$ be a real open interval, and let $E$ be any real or complex Banach space. Let the function $f$ in $L^1(J,E)$ be such that $\int_J(\varphi\,f)=0_E$ holds for all compactly supported smooth $\varphi:J\to\bbR$. Then $f(t)=0_E$ holds for almost all $t\in J$.
Proof. Let $x\mapsto\|x\|$ be a norm for $E$. Having $f$ a.e. the limit of a sequence of simple functions, there are a Lebesgue null set
$N_1\inc J$ and a separable closed linear subspace $S$ in $E$ such that $f(t)\in S$ holds for all $t\in J\setminus N_1$. By Lemma 8.15.1 (p. 573) in R. E. Edward's Functional Analysis there is a countable set $D$ in the unit ball of the dual of $E$ such that
$\|x\|=\sup\{|u(x)|:u\in D\}$ holds for all $x\in S$. By classical results, it follows existence of a Lebesgue null set $N_0\inc J$ such that $u\circ f(t)=0$ holds for all $t\in J\setminus N_0$ and $u\in D$. It follows that $\|f(t)\|=0$ holds for all $t\in J\setminus N_0$.
I do not know whether there is a published reference where the above Lemma would be explicitly stated or proved.
