I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question:
Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f \colon [a,b] \longrightarrow X$ a function. How can we integrate such a function?
I could already find an answer with the $\textbf{Riemann Integral for Banach space-valued functions}$ (which is quite similar to the comon Riemann Integral) and the $\textbf{Bochner Integral}$ (which is similar to the Lebesgue Integral).
But so far I only know some theoretical results about those integrals (only the basical ones) and I have not yet seen or calculated a practical example.
Now I wonder if anybody could present me different examples of such a integral. (I am looking for such nice and epical integrals we know from Complex analysis or we could calculate using an $d$-dimensional Spherical coordinate system or something similar.)
I am also looking for any kind of (nice) calculations involving Integration in Banach Spaces.
If anybody knows a rewarding (not too hard) theorem/proof involving Integration in Banach Spaces this would also interest me.
I hope you understand what I am searching for…
Best Answer
One useful example is the holomorphic functional calculus. It allows us to generalize Cauchy's integral formula from complex analysis in one variable to evaluate functions of operators.
Let $V$ be a Banach space and let $T$ be a bounded linear operator on $V$. If $\Gamma$ is a positively oriented rectifiable Jordan curve such that the spectrum of $T$ is contained in the interior of $\Gamma$, then for each function $f$ holomorphic on and inside $\Gamma$,
$$ f(T) = \frac{1}{2 \pi i} \oint_{\Gamma} f(\zeta) (\zeta I - T)^{-1} \, dz $$
The integrand is a function whose arguments are in $\mathbb{C}$ and that takes values in $V$, and hence it requires Bochner integration to make well-defined. The above formula is the proper generalization of the Cauchy integral formula
$$ f(z) = \frac{1}{2 \pi i} \oint_{\Gamma} f(\zeta) (\zeta - z)^{-1} \, dz,$$
where $\Gamma$ encloses $z$ (the value $z$ being the only element in the spectrum of the map $x \mapsto zx$).
This formula allows you to derive Bochner integral formulations for expressions like $\exp(T)$ or $\log(T)$ for certain linear operators $T$. In the case that $V = \mathbb{C}^{n \times n}$, then $T$ is a matrix and the Cauchy integral formulation for $\exp(T)$ matches the regular definition of the matrix exponential.