Continuity of $$L_{p}$$ Balls and an Application to Input-Output Systems
Anar Huseyin, Nesir Hüseyin, Kh. G. Guseinov · Mathematical Notes · 2022
In this paper, the continuity of the set-valued map $$p\rightarrow B_{\Omega,\mathcal{X},p}(r)$$ , $$p\in (1,+\infty)$$ , is proved where $$B_{\Omega,\mathcal{X},p}(r)$$ is the closed ball of radius $$r$$ in the space $$L_{p}(\Omega,\Sigma,\mu; \mathcal{X})$$ centered at the origin, $$(\Omega,\Sigma,\mu)$$ is a finite and positive measure space, and $$\mathcal{X}$$ is a separable Banach space. An application to input-output systems described by Urysohn type integral operators is discussed.