Locally convex spaces

Horváth, J. I.

doi:10.1007/bfb0068227

Cited by 17 publications

(29 citation statements)

References 31 publications

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“…“⇒”: Let

(\Omega _k)_{k \in \mathbb {N}}

be a locally finite open cover of Ω by relatively compact subsets

\Omega _k

. Then, there exist functions

\varphi _k \in \mathcal {D}(\Omega )

\varphi _k(x) \ge 0

for all

x \in \Omega

such that

\operatorname{supp}\varphi _k \subset \Omega _k

, the family

(\operatorname{supp}\varphi _k)_{k \in \mathbb {N}}

is locally finite, and

\sum _{k=1}^\infty \varphi _k(x) = 1

x \in \Omega

[26, Theorem 4, p. 168]. By defining

a(k) = 1 + \sup _{f \in B} \left\Vert \varphi _k \cdot f\right\Vert _q

, we see that the function

R^{n} \to double-struckR, x \mapsto \sum k =

…”

Section: Bounded Sets In the Fréchet Spaces Llocq$l^{q}_{\text {Loc}}...mentioning

confidence: 99%

Projective descriptions of spaces of functions and distributions

Bargetz

Nigsch

Ortner

2023

Mathematische Nachrichten

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We present projective descriptions of classical spaces of functions and distributions. More precisely, we provide descriptions of these spaces by seminorms, which are defined by a combination of classical norms and multiplication or convolution with certain functions. These seminorms are simpler than the ones given by a supremum over bounded or compact sets. K E Y W O R D Sbounded sets, compact sets, distribution spaces, 𝑝-integrable smooth functions, topology of uniform convergence on bounded sets, topology of uniform convergence on compact setsThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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“…“⇒”: Let

(\Omega _k)_{k \in \mathbb {N}}

be a locally finite open cover of Ω by relatively compact subsets

\Omega _k

. Then, there exist functions

\varphi _k \in \mathcal {D}(\Omega )

\varphi _k(x) \ge 0

for all

x \in \Omega

such that

\operatorname{supp}\varphi _k \subset \Omega _k

, the family

(\operatorname{supp}\varphi _k)_{k \in \mathbb {N}}

is locally finite, and

\sum _{k=1}^\infty \varphi _k(x) = 1

x \in \Omega

[26, Theorem 4, p. 168]. By defining

a(k) = 1 + \sup _{f \in B} \left\Vert \varphi _k \cdot f\right\Vert _q

, we see that the function

R^{n} \to double-struckR, x \mapsto \sum k =

…”

Section: Bounded Sets In the Fréchet Spaces Llocq$l^{q}_{\text {Loc}}...mentioning

confidence: 99%

Projective descriptions of spaces of functions and distributions

Bargetz

Nigsch

Ortner

2023

Mathematische Nachrichten

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“…In the particular case where (X, d X (•, •)) exhibits a linear structure -e.g. when it is a Banach or a Fréchet space, see for instance [63] -, we define the closed convex hull of one of some of its subsets B ⊂ X as…”

Section: Set-valued Analysis and Topological Properties Of The Spacementioning

confidence: 99%

Viability and Exponentially Stable Trajectories for Differential Inclusions in Wasserstein Spaces

Bonnet

Frankowska

2022

2022 IEEE 61st Conference on Decision and Control (CDC)

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In this article, we derive necessary and sufficient conditions for the existence of solutions to stateconstrained continuity inclusions in Wasserstein spaces whose right-hand sides may be discontinuous in time. These latter are based on fine investigations of the infinitesimal behaviour of the underlying reachable sets, through which we show that up to a negligible set of times, every admissible velocity of the inclusion can be approximately realised as the metric derivative of a solution of the dynamics, and vice versa. Building on these results, we are able to establish necessary and sufficient geometric conditions for the viability and invariance of stationary and time-dependent constraints, which involve a suitable notion of contingent cones in Wasserstein spaces, presented in ascending order of generality. We then close the article by exhibiting two prototypical examples of constraints sets appearing in applications for which one can compute relevant subfamilies of contingent directions.

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“…The following result due to Dieudonné and Schwartz (see [7], page 308) is needed. S(Of ) = S(T * f ) = (S * T )(f ).…”

Section: Approximation and Existence Theoremsmentioning

confidence: 99%