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 X {\displaystyle X} be a topological Hausdorff space, a (continuous) Banach bundle over X {\displaystyle X} is a tuple B = ( B , π ) {\displaystyle {\mathfrak {B}}=(B,\pi )} , where B {\displaystyle B} is a topological Hausdorff space, and π : B → X {\displaystyle \pi \colon B\to X} is a continuous, open surjection, such that each fiber B x := π − 1 ( x ) {\displaystyle B_{x}:=\pi ^{-1}(x)} is a Banach space. Which satisfies the following conditions:
The map b ↦ ‖ b ‖ {\displaystyle b\mapsto \|b\|} is continuous for all b ∈ B {\displaystyle b\in B}
The operation + : { ( b 1 , b 2 ) ∈ B × B : π ( b 1 ) = π ( b 2 ) } → B {\displaystyle +\colon \{(b_{1},b_{2})\in B\times B:\pi (b_{1})=\pi (b_{2})\}\to B} is continuous For every λ ∈ C {\displaystyle \lambda \in \mathbb {C} } , the map b ↦ λ ⋅ b {\displaystyle b\mapsto \lambda \cdot b} is continuous If x ∈ X {\displaystyle x\in X} , and { b i } {\displaystyle \{b_{i}\}} is a net in B {\displaystyle B} , such that ‖ b i ‖ → 0 {\displaystyle \|b_{i}\|\to 0} and π ( b i ) → x {\displaystyle \pi (b_{i})\to x} , then b i → 0 x ∈ B {\displaystyle b_{i}\to 0_{x}\in B} , where 0 x {\displaystyle 0_{x}} denotes the zero of the fiber B x {\displaystyle B_{x}} . If the map b ↦ ‖ b ‖ {\displaystyle b\mapsto \|b\|} is only upper semi-continuous, B {\displaystyle {\mathfrak {B}}} is called upper semi-continuous bundle.
Examples
Trivial bundle Let A be a Banach space, X be a topological Hausdorff space. Define B := A × X {\displaystyle B:=A\times X} and π : B → X {\displaystyle \pi \colon B\to X} by π ( a , x ) := x {\displaystyle \pi (a,x):=x} . Then ( B , π ) {\displaystyle (B,\pi )} is a Banach bundle, called the trivial bundle
See also Banach bundles in differential geometry
References