If we define the (Riemann) integral of an abstract function, i.e. a function $f:[a,b]\to Y$ where $Y$ is a Banach space, as$$\int_a^b F(t)dt:=\lim_{\max(t_{k+1}-t_k)\to 0}\sum_{k=0}^{n-1}(t_{k+1}-t_k)F(\xi_k)$$where $a=t_0<t_1<\ldots<t_n=b$ is one of the partitions whose intervals' maximum length approaches $0$, I read two unproved statement in Kolmogorov-Fomin's (p. 486 here) which I find very interesting:
- If $F$ is continuous, then it is integrable. The text says that the proof is analogous to the case $F:[a,b]\to\mathbb{R}$, but all the proofs I have found for the real-valued case rely on Darboux sums where maxima and minima and other properties of the real numbers are used, while we have values in a general Banach space, here.
- If $U:Y\to Z$ is a linear map between two Banach spaces then $\int_a^b UF(t)dt=U\int_a^bF(t)dt$ for any integrable $F$.
I have searched a lot in the web, but I find no proofs of these statements. Could anybody write a proof, the most elementary possible, since Kolmogorov-Fomin's explains very few things about such integral, or give a link to one? I thank you for any answer!
As one commenter pointed out, in The Integral by Krantz, to show that continuous functions into a Banach space $X$ are integrable, we can essentially mimic the proof given for real-valued functions.
More precisely, we first define the integral of $f:I\to X$ by declaring that if there exists an element $I\in X$ such that for each $\varepsilon>0$ and each partition $\mathcal P$ of $I$, there is a $\delta>0$ such that whenever the mesh of the partition $\mathcal P$ is $<\delta$, we have that $\lVert \mathcal R(f,\mathcal P) - I\rVert < \varepsilon$. We declare that $I = \int_a^b f(t)\,dt$.
This is a good definition because the norm on Banach spaces makes them into metric spaces, so if $I$ exists, it is unique.
Now if $f$ is continuous, we can prove it is integrable mimicking the proof for real-valued functions by showing that $\{\mathcal R(\mathcal P_n)\}$ is a Cauchy sequence for some appropriate choice of partitions with $\operatorname{mesh}\mathcal P_n\to 0$ as $n\to\infty$.
A corollary of this is (cf. Pugh's book Real Mathematical Analysis)
$C^1$ Mean Value Theorem If $U\subset \mathbb R^n$ is open, and $f:U\to\mathbb R^m$ is of class $C^1$, so the map $Df:\mathbb R^n\to \mathcal L(\mathbb R^n,\mathbb R^m)$ exists and is continuous, and if the segment $[p,q]$ is contained in $U$ then $$ f(q) - f(p) = T(q-p). $$ where $T$ is the average derivative of $f$ on the segment, $$ T = \int_0^1(Df)_{p+t(q-p)}\,dt. $$ The proof is the fact that $\mathcal L(\mathbb R^n,\mathbb R^m)$ is a Banach space and an application of our corollary.