Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces

TL;DR

This paper develops advanced infinite-dimensional differential calculus techniques and applies them to construct new vector bundles and analyze Poisson vector spaces, ensuring continuity of key structures.

Contribution

It introduces new results in infinite-dimensional calculus and applies them to construct vector bundles and establish continuity in Poisson vector spaces.

Findings

Construction of dual bundles, tensor products, and direct sums.

Proof of continuity of the Poisson bracket.

Continuity of Hamiltonian vector fields.

Abstract

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, like dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses). 2. in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow hypocontinuity of bilinear mappings to be exploited. Notably, we encounter $k_{{\mathbb R}}$-spaces and locally convex spaces $E$ such that $E\times E$ is a $k_{{\mathbb R}}$-space.