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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...

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There's nothing fundamentally new in terms of the calculations. Any reasonable definition of a Banach space-valued integral should have the property that if $v^{\ast} : B \to \mathbb{R}$ is a linear functional, then $v^{\ast} (\int f) = \int v^{\ast} f$, where the latter is an ordinary one-dimensional integral. – Qiaochu Yuan Apr 6 '14 at 5:02
Is this a soft version of Hille's Theorem? If so, where can I find a proof for it? I already studied the proof of Hille's Theorem, but I need some more time because you need a lot of measure theory for it. – user140335 Apr 6 '14 at 13:24

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.

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Very nice useful example! I can need this one. Can I use your statement also for "toy contours" as they are defined in "COMPLEX ANALYSIS - Stein & Shakarchi"? – user140335 Apr 4 '14 at 23:46
This material is used, among other places, in some of the foundational results leading up to the proof of the Gelfand-Naimark theorem; see… for some indications of how this works (I black-boxed the complex analysis though). – Qiaochu Yuan Apr 6 '14 at 5:05

First of all, in order to "calculate" something explicitly you need a reasonably nice Banach space $B$ as your target. For instance, suppose your Banach space is $B=C([a,b])$. Then a continuous function $f: [0,T]\to B$ is nothing but a continuos function of two variables $F(x,t)$, $x\in [a,b], t\in [0,T]$: $$ f(t)(x)= F(x,t). $$ Now, computing the Bochner integral $$ \int_{0}^T f(t)dt $$ simply amounts to (if you follow the definition) computing the integral $$ \int_{0}^T F(x,t)dt $$ which is something you surely saw in a calculus of several variables class.

You can use a similar computation if your target is, say, $L^p([a,b])$ and so on.

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