Banach bundle (non-commutative geometry)


In mathematics, a Banach bundle is a fiber bundle over a topological Hausdorff space, such that each fiber has the structure of a Banach space.

Definition

Let be a topological Hausdorff space, a Banach bundle over is a tuple, where is a topological Hausdorff space, and is a continuous, open surjection, such that each fiber is a Banach space. Which satisfies the following conditions:
  1. The map is continuous for all
  2. The operation is continuous
  3. For every, the map is continuous
  4. If, and is a net in, such that and, then, where denotes the zero of the fiber.
If the map is only upper semi-continuous, is called upper semi-continuous bundle.

Examples

Trivial bundle

Let A be a Banach space, X be a topological Hausdorff space. Define and by. Then is a Banach bundle, called the '''trivial bundle'''