I've recently started a course on PDEs, and one of the first constructions we made was the following.
Say we have a PDE: $$F(t,x_1,\dots,x_n,u,u_t,u_{x_1},\dots,u_{x_n},\dots) = 0$$
with $u(t,x): \Bbb R\times \Bbb R^{n}\to\Bbb R$ being as smooth as necessary. For my question assume $n = 1$ for simplicity. Now we interpret $u(t,x$) as $u(t)(x)$ with $$\begin{align}u:\Bbb R& \to \mathcal C^h(\mathbb R)& \\ \quad \qquad\qquad t\mapsto &\quad\quad u(t):\Bbb R\to\Bbb R \\ & \quad\qquad x\mapsto u(t)(x) = u(t,x)\end{align}$$
(yes, I use the same $u$ for different functions, but I think which is which will be clear from context in the following).
Then, PDEs could be interpreted as $$u'(t) = H(u)$$ with $H :\mathcal C^h(\Bbb R)\to \mathcal C^h(\Bbb R)$, and the derivative of a function whose values are in $\mathcal C^h(\Bbb R)$ is defined with the usual difference quotient (and convergence in the $\mathcal C^h $ norm). This allowed for some developments on existence, uniqueness, etc.
My problem is that we didn't establish, or even mention the obviously desired correspondence: $$(u'(t))(x) = u_t(t,x)$$ What's more, we went through many deductions that involved passing to integral form, and here again, you obviously want $$\left (\int_a^b u(s)\, ds \right )(x) =\int_a^b u(s,x)\,ds$$
along with $u$ (with values in the function space) to be measurable in the first place (and with which measure??).
In summary, I feel that these sort of things should be established, if they are to help discuss PDEs, right? Please let me know if my question isn't clear. If it is, could you refer me to a book or other material that discusses this?