  The Bochner integral is a generalization of the Lebesgue integral, for
functions taking their values in a Banach space. Therefore, both its
mathematical definition and its formalization in the Coq proof assistant are
more challenging as we cannot rely on the properties of real numbers. Our
contributions include an original formalization of simple functions, Bochner
integrability defined by a dependent type, and the construction of the proof of
the integrability of measurable functions under mild hypotheses (weak
separability). Then, we define the Bochner integral and prove several theorems,
including dominated convergence and the equivalence with an existing
formalization of Lebesgue integral for nonnegative functions.
