Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces

Abstract

We prove various results in infinite-dimensional differential calculus that relate the 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, such as 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 the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

On the geometry of diffeological vector pseudo-bundles and infinite dimensional vector bundles: automorphisms, connections and covariant derivatives

“…A second problem that we have to highlight is the problem of smoothness of the evaluation of linear maps. Following the review [17], when F is a locally convex vector space, and unless F is a Banach space, the evaluation map L(F ) × F → F is merely hypocontinuous. This leads to the definition of infinite dimensional vector bundle with structure group where a prescribed Lie group G with Lie algebra g is such that G and g are acting smoothly on F. With this assumption, the construction of a principal bundle of G−frames, a bundle of G−endomorphisms, a space of g−valued connection 1-forms and their G−covariant derivatives are safely defined, see e.g.…”