Smoothing operators for vector-valued functions and extension operators

Abstract

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.

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

“…Then, N q := {y ∈ E : q(y) = 0} is a closed vector subspace of E and y + N q q := q(y) for y ∈ E defines a norm on E q := E/N q making the map α q : E → E q , y → y + N q continuous linear. By (12), we have…”

“…In the preceding situation, i∈I E i is an equivariant L-vector bundle of class C r K over M. Remark 12. If M is a C r R -manifold, then every x ∈ M has an open neighbourhood U which is C r R -diffeomorphic to a convex open subset W in the modelling space Z of M. If W can be chosen C r R -paracompact, then every C r R -vector bundle over U is trivialisable (see [12] (Corollary 15.10)). The latter condition is satisfied, for example, if Z is finite-dimensional, a Hilbert space, or a countable direct limit of finite-dimensional vector spaces (and hence a nuclear Silva space), cf.…”

“…We show that the Poisson bracket associated with a continuous Lie bracket is always continuous (Theorem 1) and that the linear map C ∞ (E, R) → C ∞ (E, E) taking a smooth function to the associated Hamiltonian vector field is continuous (Theorem 2). Ideas from [8] and the current article were also taken further in [12] (Section 13).…”

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

“…R -paracompact, then every C r R -vector bundle over U is trivializable (see [25,Corollary 15.10]). The latter condition is satisfied, for example, if Z is finite dimensional, a Hilbert space, or a countable direct limit of finite-dimensional vector spaces (cf.…”

Manifolds of mappings on cartesian products

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