Continuity of $L_p$ Balls and an Application to Input-Output Systems

Abstract

In this paper, the continuity of the set-valued map $p\rightarrow B_{\Omega, \mathcal{X},p}$, $p\in (1,+\infty)$, is proved where $B_{\Omega, \mathcal{X},p}$ 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.