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I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not need to worry about measurability issues. Due to presence of the locally trivializing map in their construction, given a connected base space, all the fibers are isomorphic to each other. Therefore, fibers with varying dimensions are not allowed.

Other authors, like Dautray and Lions, or Birman and Solomjak define measurable Hilbert bundles and do not seem to insist on isomorphic fibers.

My question is, in the case of isomorphic fibers, why not simply use Bochner spaces like $C(X;L^2(\Omega))$? Reed and Simon (in the fourth volume) insist that the focus is on the fibers rather than on $X$ and promise to cover general Hilbert bundles in Chapter XVI of their series "Methods of Modern Mathematical Physics", but as far as I know, that chapter never appeared.

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not need to worry about measurability issues. Due to presence of the locally trivializing map in their construction, given a connected base space, all the fibers are isomorphic to each other. Therefore, fibers with varying dimensions are not allowed.

Other authors, like Dautray and Lions, or Birman and Solomjak define measurable Hilbert bundles and do not seem to insist on isomorphic fibers.

My question is, in the case of isomorphic fibers, why not simply use Bochner spaces like $C(X;L^2(\Omega))$? Reed and Simon (in the fourth volume) insist that the focus is on the fibers rather than on $X$ and promise to cover general Hilbert bundles in Chapter XVI of their series "Methods of Modern Mathematical Physics", but as far as I know, that chapter never appeared.

Some other questions:

1. Is the "constant rank" condition necessary in order to put a differentiable (or continuous) structure on the bundle?

2. In contrast, is it true that the "constant rank" condition is not imposed on measurable Hilbert bundles because measurability is a weaker condition that allows for variation of fibers?

3. Are "constant rank" fibrations more "natural" in some way in mathematics?

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not need to worry about measurability issues. Due to presence of the locally trivializing map in their construction, given a connected base space, all the fibers are isomorphic to each other. Therefore, fibers with varying dimensions are not allowed.

Other authors, like Dautray and Lions, or Birman and Solomjak define measurable Hilbert bundles and do not seem to insist on isomorphic bundles.

My question is, in the case of isomorphic fibers, why not simply use Bochner spaces like $C(X;L^2(\Omega))$? Reed and Simon (in the fourth volume) insist that the focus is on the fibers rather than on $X$ and promise to cover general Hilbert bundles in Chapter XVI of their series "Methods of Modern Mathematical Physics", but as far as I know, that chapter never appeared.

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not need to worry about measurability issues. Due to presence of the locally trivializing map in their construction, given a connected base space, all the fibers are isomorphic to each other. Therefore, fibers with varying dimensions are not allowed.

Other authors, like Dautray and Lions, or Birman and Solomjak define measurable Hilbert bundles and do not seem to insist on isomorphic fibers.

My question is, in the case of isomorphic fibers, why not simply use Bochner spaces like $C(X;L^2(\Omega))$? Reed and Simon (in the fourth volume) insist that the focus is on the fibers rather than on $X$ and promise to cover general Hilbert bundles in Chapter XVI of their series "Methods of Modern Mathematical Physics", but as far as I know, that chapter never appeared.