Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces
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 construc...
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MDPI AG
2022-05-01
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| author | Helge Glöckner |
| author_facet | Helge Glöckner |
| author_sort | Helge Glöckner |
| collection | DOAJ |
| description | 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 <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>k</mi><mi mathvariant="double-struck">R</mi></msub></semantics></math></inline-formula>-spaces and locally convex spaces <i>E</i> such that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>×</mo><mi>E</mi></mrow></semantics></math></inline-formula> is a <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>k</mi><mi mathvariant="double-struck">R</mi></msub></semantics></math></inline-formula>-space. |
| first_indexed | 2024-03-10T03:20:32Z |
| format | Article |
| id | doaj.art-8321ba517ae84974b92d18e3e67aecf0 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T03:20:32Z |
| publishDate | 2022-05-01 |
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| series | Axioms |
| spelling | doaj.art-8321ba517ae84974b92d18e3e67aecf02023-11-23T10:04:16ZengMDPI AGAxioms2075-16802022-05-0111522110.3390/axioms11050221Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector SpacesHelge Glöckner0Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, GermanyWe 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 <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>k</mi><mi mathvariant="double-struck">R</mi></msub></semantics></math></inline-formula>-spaces and locally convex spaces <i>E</i> such that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>×</mo><mi>E</mi></mrow></semantics></math></inline-formula> is a <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>k</mi><mi mathvariant="double-struck">R</mi></msub></semantics></math></inline-formula>-space.https://www.mdpi.com/2075-1680/11/5/221vector bundledual bundledirect sumcompletiontensor productcocycle |
| spellingShingle | Helge Glöckner Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces Axioms vector bundle dual bundle direct sum completion tensor product cocycle |
| title | Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces |
| title_full | Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces |
| title_fullStr | Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces |
| title_full_unstemmed | Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces |
| title_short | Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces |
| title_sort | aspects of differential calculus related to infinite dimensional vector bundles and poisson vector spaces |
| topic | vector bundle dual bundle direct sum completion tensor product cocycle |
| url | https://www.mdpi.com/2075-1680/11/5/221 |
| work_keys_str_mv | AT helgeglockner aspectsofdifferentialcalculusrelatedtoinfinitedimensionalvectorbundlesandpoissonvectorspaces |