Differential calculus on multiple products

Alzaareer, Hamza

doi:10.1016/j.indag.2019.07.008

Cited by 12 publications

(25 citation statements)

References 11 publications

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“…N on a product of smooth manifolds with rough boundary is C β,α with α ∈ (N 0 ∪ {∞}) n and β ∈ (N 0 ∪ {∞}) m , then the map [1,Lemma 3.3]).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…is C α . The latter then holds for any such charts, by the chain rule for C α -maps (as in [1,Lemma 3.16]).…”

Section: 12mentioning

confidence: 92%

“…We work in the setting of differential calculus going back to Andrée Bastiani [5] (see [10,15,16,[21][22][23] for discussions in varying generality), also known as Keller's C k c -theory [19]. For C α -maps, see [1] (cf. [3] and [15] for the case of two variables, α ∈ (N 0 ∪ {∞}) 2 ).…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…In [1], exponential laws were provided for function spaces on products of pure manifolds. The one we need remains valid for manifolds which need not be pure: Lemma 3.13 Let N 1 , .…”

Section: Lemma 35 Using Compact-open C α -Topologies We Havementioning

confidence: 99%

“…product of smooth manifolds (possibly with rough boundary) modeled on locally convex spaces and f : 4 Note that due to Lemma 4.3 (a), the evaluation on a canonical manifold is a C ∞,α -map whence it is at least continuous. For a C k -manifold M which is C k -regular 1 and a locally convex space E = {0}, it is well known that for the compact-open C k -topology the evaluation ev : We now turn to smoothness properties of the composition map.…”

Section: Lemma 43 If C α (M Nmentioning

confidence: 99%

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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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“…N on a product of smooth manifolds with rough boundary is C β,α with α ∈ (N 0 ∪ {∞}) n and β ∈ (N 0 ∪ {∞}) m , then the map [1,Lemma 3.3]).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…is C α . The latter then holds for any such charts, by the chain rule for C α -maps (as in [1,Lemma 3.16]).…”

Section: 12mentioning

confidence: 92%

Section: Preliminaries and Notationmentioning

confidence: 99%

“…In [1], exponential laws were provided for function spaces on products of pure manifolds. The one we need remains valid for manifolds which need not be pure: Lemma 3.13 Let N 1 , .…”

Section: Lemma 35 Using Compact-open C α -Topologies We Havementioning

confidence: 99%

Section: Lemma 43 If C α (M Nmentioning

confidence: 99%

See 3 more Smart Citations