---
tag: 9A5F3JGTC0
kind: concept
name: Pettis integral
msc: [28B05, 46G10]
sources:
  - url: https://en.wikipedia.org/wiki/Pettis_integral
same_as:
  wikidata: Q211188
  wikipedia: Pettis integral
---

## Summary

The Pettis integral of a function with values in a Banach space is a vector in that space.
Each continuous linear functional maps it to the integral of the corresponding scalar function.
It is a weak notion of integral, defined through the dual space.
Every Bochner integrable function is Pettis integrable with the same integral, but in infinite dimensions there are Pettis integrable functions that are not Bochner integrable.
It is also called the Gelfand–Pettis integral.

## Explanations

### L1 human

To integrate a vector-valued function, one can integrate each coordinate.
In an infinite-dimensional space, the role of coordinates is played by continuous linear functionals.
The Pettis integral is a vector that has the right value in every such coordinate, for every measurable set.
For a function into $\R^2$, it is the vector whose two coordinates are the integrals of the two components of the function.