Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus

2020

Preprint

Let Ω ⊆ R d be open and ℓ ∈ N0 ∪ {∞}. Given a locally convex topological vector space F , endow C ℓ (Ω, F ) with the compact-open C ℓtopology. For ℓ < ∞, we describe a sequence (Sn) n∈N of continuous linear operators Sn :Moreover, we study the existence of continuous linear right inverses for restriction maps C ℓ (R d , F ) → C ℓ (R, F ), γ → γ|R for subsets R ⊆ R d with dense interior. As an application, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.12 As functions f with domain C ℓ (R, F ) are a frequent topic in infinite-dimensional calculus, we now denote the elements of C ℓ (R, F ) by γ (rather than f ).13 Let P := lim ←−C n (R, F ) ⊆ n∈N 0 C n (R, F ) be the standard projective limit. One readily verifies that the linear map φ : C ∞ (R, F ) → P , γ → (γ) n∈N 0 is bijective, continuous, and that φ −1 is continuous.

Section: Recall That a Hausdorff Topological Spacesupporting

confidence: 52%

mentioning

confidence: 75%

See 1 more Smart Citation

Smoothing operators for vector-valued functions and extension operators

2020

Preprint

“…Applications related to locally convex Poisson vector spaces. In the wake of works by Odzijewicz and Ratiu on Banach-Poisson vector spaces and Banach-Poisson manifolds [47,48], certain locally convex Poisson vector spaces were introduced [20], which generalize the Lie-Poisson structure on the dual space of a finite-dimensional Lie algebra going back to Kirillov, Kostant and Souriau. By now, the latter spaces can be embedded in a general theory of locally convex Poisson manifolds (see [44]; for generalizations of finitedimensional Poisson geometry with a different thrust, cf.…”

Section: Introductionmentioning

confidence: 99%

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

2022

Preprint

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 R -spaces and locally convex spaces E such that E × E is a k R -space.

“…See, e.g.,[12, Proposition 16.8] for the equivalence of this definition with more classical ones (cf. also Proposition 4 in [6, Chapter III, §5, no.…”

mentioning

confidence: 99%

Upper bounds for continuous seminorms and special properties of bilinear maps

Topology and its Applications

2012

Self Cite

If E is a locally convex topological vector space, let (P (E), ) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for θ an infinite cardinal those locally convex spaces E which have the θ-neighbourhood property introduced by E. Jordá, meaning that all sets M of continuous seminorms of cardinality |M | ≤ θ have an upper bound in P (E). On the other hand, we study bilinear maps β : E 1