Integration in Banach spaces - interesting, nice and non-trivial examples needed

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.

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.

Elaborating a bit on @QiaochuYuan's comment: for almost every purpose, we certainly want the property that $T\int f=\int Tf$, where $f$ is $V$-valued, and $T$ is a continuous linear map $V\to W$. Under very mild hypotheses on the spaces $V,W$ (locally convex, quasi-complete suffices), by Hahn-Banach the desired property already follows from requiring $\lambda\int f=\int \lambda\circ f$ for continuous linear functionals $\lambda$. (We would also want estimates on $\int f$, but/and that does/will come out of one of several constructions.) This characterization is often called a Gelfand-Pettis integral, or weak integral (not because the conclusion is weak, but because the hypothesis is weak).

The condition with linear functionals already implies (by Hahn-Banach) that if there exists any vector $v=\int f\in V$ at all, it is unique. There's no ambiguity in what it is... if it exists.

Not all uniquely characterized things exist, as it turns out, so proof of existence (under various hypotheses) is important. Constructions suffice, and may prove useful incidental properties.

This set-up already allows the Schwartz-Grothendieck vector-valued extension of Cauchy-Goursat theory of complex functions in one variable.

Many basic purposes are fulfilled by looking at Banach-space-valued functions, perhaps continuous with compact support. But in practice we often want the weaker "strong" topology on operators. And we may want test-function-valued functions (not Frechet, but strict-colimit-of-Frechet), or distribution-valued. A small amount of work does verify that all spaces $V$ are "quasi-complete" (and locally convex), so that compactly-supported, continuous, $V$-valued functions (as a useful example) have integrals. In a very positive sense, we can prove that the situation is fairly "idiot-proof", although understanding that point is considerably more complicated than simply using the ideas in blissful naivete. :)