I want to learn more about the fundamental theorem of curves. Wikipedia states the theorem for ${\bf R}^3$ only but I found another source (Theorem 5.5.18, in German only) where it is proved for arbitrary curves in Banach Spaces, using Riemann integration. It says:
Given $f:[a,b]\rightarrow E$ continuous and setting $\phi(x):=\int^x_a f(t)dt$ then $\phi$ is differentiable with $\phi'=f$. If on the other hand $f:[a,b]\rightarrow E$ is differentiable with Riemann integrable derivation, then $\int^b_a f'(t)dt=f(b)-f(a)$.
I just wondered if there is a more general setting, using Pettis integral or Bochner integral, or are those definitions of the integral all the same if the measure space is $[0,1]$?
Also, are there other recommended books / online resources which deal with the fundamental theorem of curves in Banach spaces?
Thanks!
Edit: I just read here that Bochner is weaker than Riemann in the setting of curves, whereas Pettis is (strictly?) stronger. But I'm also interested if one can weaken the premises of the theorem, e.g. using absolutely continuous instead of continuously differentiable or integrable instead of continuous.
I think I got some answer for the second part of the theorem. If $f:[a,b]\rightarrow E$ is absolutely continuous and differentiable almost everywhere, then for $\phi\in E^*$ we have $\phi\circ f$ absolutely continuous hence the fundamental theorem for real valued functions gives us $$\int^b_a (\phi\circ f)'=\phi(f(b))-\phi(f(a)).$$ Combining this with $(\phi\circ f)'=\phi\circ f'$ almost everywhere and the linearity of $\phi$ we have $$\int^b_a \phi\circ f'=\phi(f(b)-f(a))$$ but since $\phi\in E^*$ was arbitrary this gives exactly the definition of the Pettis integral, hence $f$ is Pettis integrable and fulfills $$\int^b_a f'=f(b)-f(a).$$ I'm still unsure if one needs to assume $f$ being differentiable almost everywhere, but I could imagine that absolute continuity does not imply differentiability almost everywhere in Banach spaces as it does for the real numbers.