Using partitions of the unity ($(PU)$-partition), a new definition of an integral is given for a function f : [a, b] \to X, where X is a Banach space, and it is proved that this integral is equivalent to the Bochner integral.
@article{37, title = {An integral for a Banach valued function}, journal = {Tatra Mountains Mathematical Publications}, volume = {43}, year = {2009}, doi = {10.2478/tatra.v44i0.37}, language = {EN}, url = {http://dml.mathdoc.fr/item/37} }
Riccobono, Giuseppa. An integral for a Banach valued function. Tatra Mountains Mathematical Publications, Tome 43 (2009) . doi : 10.2478/tatra.v44i0.37. http://gdmltest.u-ga.fr/item/37/