A -seeley-extension-theorem for Bastiani’s differential calculus

Hanusch, Maximilian

doi:10.4153/s0008414x21000596

Cited by 2 publications

(2 citation statements)

References 19 publications

(42 reference statements)

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“…Remark 1 • It is immediate from Lemma 1 that the definition made in (12) does not depend on the explicit choice of the fundamental system H ⊆ Sem(F). • Since we will make use of the differentiability results obtained in [8], we explicitly mention that in [8] an equivalent definition of Mackey convergence (more suitable for the technical argumentation there) was used.…”

Section: Mackey Convergencementioning

confidence: 99%

The Lax equation and weak regularity of asymptotic estimate Lie groups

Hanusch¹

2023

Ann Glob Anal Geom

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We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor’s sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker–Campbell–Dynkin–Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context.

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Section: Mackey Convergencementioning

confidence: 99%

The Lax equation and weak regularity of asymptotic estimate Lie groups

Hanusch¹

2023

Ann Glob Anal Geom

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“…If U may not be open, but has dense interior U o and is locally convex in the sense that each x ∈ U has a convex neighborhood in U , following [15] a map f : [13], cf. [17]).…”

Section: Consider Locally Convex Spaces E F and A Mapmentioning

confidence: 99%

Manifolds of mappings on Cartesian products

Glöckner

Schmeding

2022

Ann Glob Anal Geom

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Given smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\times \cdots \times M_n\rightarrow N$$ f : M 1 × ⋯ × M n → N which are $$C^\alpha $$ C α in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $$\le \alpha _j$$ ≤ α j in the jth variable for $$j\in \{1,\ldots , n\}$$ j ∈ { 1 , … , n } , in local charts. We show that $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) admits a canonical smooth manifold structure whenever each $$M_j$$ M j is compact and N admits a local addition. The case of non-compact domains is also considered.

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A -seeley-extension-theorem for Bastiani’s differential calculus

Cited by 2 publications

References 19 publications

The Lax equation and weak regularity of asymptotic estimate Lie groups

The Lax equation and weak regularity of asymptotic estimate Lie groups

Manifolds of mappings on Cartesian products

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