In the book Modern Fourier Analysis by Grafakos in the proof of the T(1) Theorem in the implication that shows $L^2$ boundedness (in the 3rd edition this is on page 243), Grafakos considers a function $(0,\infty)\to \mathcal{S}_0'$ (the dual of the Schwartz functions with zero mean) and expresses the difference of two limits as an improper integral by means of the fundamental theorem of calculus.
It is not clear to me how this has to be understood. I was able to show continuous differentiability of that function in the weak* topology but already that was, in my opinion, not clear. How the integral has to be understood is then totally unclear to me. Is this meant to be an integral in a locally convex TVS? When and how is it possible to form such an integral? Why is it clear that the fundamental theorem of calculus holds?
The use of the fundamental theorem of calculus is for $s\in(\varepsilon,\frac{1}{\varepsilon})$, i.e., for finite positive values of $s$. Everything in sight is smooth and rapidly decaying. So the proof of (4.3.16) does not use any $\mathscr{S}'$-valued functions and topology on distribution spaces. You can just do it by hand with a couple Fubinis and derivation under the integral sign for smooth functions.