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.
Best Answer
$Z^\top$ is the pre-annihilator of $Z$, i.e.,
$$Z^\top = \{x\in V\colon \langle f,x\rangle = 0\; (f\in Z)\}.$$
To see that indeed we have this identification in the case $Z$ is weak*-closed, note that we may define
Every functional $f\in (V/Z^\top)^*$ canonically defines $\Phi(f)\in V^* / Z$ by $$\langle \Phi(f) + Z, v \rangle = \langle f,v+Z^\perp\rangle \quad (v\in V, z\in Z^\top).$$