---
tag: F4RH4MZQDG
kind: concept
name: Banach bundle
msc: [58B99, 57R22, 46T05]
specializes: [CX4M8K61AB]
sources:
  - url: https://en.wikipedia.org/wiki/Banach_bundle
same_as:
  wikidata: Q4853761
  wikipedia: Banach bundle
---

## Summary

A Banach bundle is a [[vector bundle|P4DZM975BV]] whose fibres are Banach spaces, possibly infinite-dimensional, over a [[Banach manifold|9K15AMGBV9]].
The transition maps between local trivialisations must vary smoothly in the operator norm.
Tangent bundles of Banach manifolds are Banach bundles, and so are the bundles of Sobolev sections that appear in nonlinear analysis on manifolds.
Many constructions of moduli spaces present the moduli space as the zero set of a section of a Banach bundle.

## Explanations

### L1 human

A vector bundle attaches a vector space to each point of a base, such as the tangent plane at each point of a surface.
In a Banach bundle these vector spaces may be infinite-dimensional, for example spaces of functions.
Locally the bundle is a product of a piece of the base and one fixed Banach space.
The way these local products are glued must change smoothly from point to point, measured in the operator norm.