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.

Let ${\displaystyle X}$ be a topological Hausdorff space, a (continuous) Banach bundle over ${\displaystyle X}$ is a tuple ${\displaystyle \mathfrak{𝔅}=(B,\pi )}$, where ${\displaystyle B}$ is a topological Hausdorff space, and ${\displaystyle \pi :B\to X}$ is a continuous, open surjection, such that each fiber ${\displaystyle B_x:={\pi }^{-1}(x)}$ is a Banach space. Which satisfies the following conditions:

1. The map ${\displaystyle b\mapsto \Vert b\Vert }$ is continuous for all ${\displaystyle b\in B}$
2. The operation ${\displaystyle +:\{(b_1,b_2)\in B\times B:\pi (b_1)=\pi (b_2)\}\to B}$ is continuous
3. For every ${\displaystyle \lambda \in \mathbb{ℂ}}$, the map ${\displaystyle b\mapsto \lambda \cdot b}$ is continuous
4. If ${\displaystyle x\in X}$, and ${\displaystyle \{b_i\}}$ is a net in ${\displaystyle B}$, such that ${\displaystyle \Vert b_i\Vert \to 0}$ and ${\displaystyle \pi (b_i)\to x}$, then ${\displaystyle b_i\to 0_x\in B}$, where ${\displaystyle 0_x}$ denotes the zero of the fiber ${\displaystyle B_x}$.^[1]

If the map ${\displaystyle b\mapsto \Vert b\Vert }$ is only upper semi-continuous, ${\displaystyle \mathfrak{𝔅}}$ is called upper semi-continuous bundle.

Let A be a Banach space, X be a topological Hausdorff space. Define ${\displaystyle B:=A\times X}$ and ${\displaystyle \pi :B\to X}$ by ${\displaystyle \pi (a,x):=x}$. Then ${\displaystyle (B,\pi )}$ is a Banach bundle, called the trivial bundle

• Banach bundles in differential geometry

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