CINECA IRIS Institutional Research Information System
IRIS è la soluzione IT che facilita la raccolta e la gestione dei dati relativi alle attività e ai prodotti della ricerca. Fornisce a ricercatori, amministratori e valutatori gli strumenti per monitorare i risultati della ricerca, aumentarne la visibilità e allocare in modo efficace le risorse disponibili.
EnglishIn this paper we give a formula for the distance from an element f of the Banach space L_p(\mu,X), 1 ≤ p ≤ \infty---where X is a Banach space and (T, M, \mu) is a positive measure space---to the subset L_p(\mu,S) of all functions whose range is (essentially) contained in a given nonempty subset S of X. This formula is in terms of the norm in L_p(\mu) of the distance function to S that is induced by f, namely, of the scalar-valued function d_f^S which maps t \in T into the distance from f(t) to S. Indeed, the sets S for which L_p(\mu,S) is nonempty are characterized. Furthermore, for such sets the function d_f^S is proved to be in L_p(\mu), and the distance from f to L_p(\mu,S) is proved to coincide with the norm of d_f^S in L_p(\mu). This generalizes previously obtained formulae---provided in [LC], 2.10 and in [L], Theorem 5---in which the distance to L_p(\mu,S) is computed in the special case of a vector subspace S of X.
Scheda prodotto non validato
Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo
| Titolo: | Computing the distance to the functions with range in a given set in Lebesgue spaces of vector-valued functions |
| Autori: | |
| Data di pubblicazione: | 2005 |
| Rivista: | |
| Abstract: | In this paper we give a formula for the distance from an element f of the Banach space L_p(\mu,X), 1 ≤ p ≤ \infty---where X is a Banach space and (T, M, \mu) is a positive measure space---to the subset L_p(\mu,S) of all functions whose range is (essentially) contained in a given nonempty subset S of X. This formula is in terms of the norm in L_p(\mu) of the distance function to S that is induced by f, namely, of the scalar-valued function d_f^S which maps t \in T into the distance from f(t) to S. Indeed, the sets S for which L_p(\mu,S) is nonempty are characterized. Furthermore, for such sets the function d_f^S is proved to be in L_p(\mu), and the distance from f to L_p(\mu,S) is proved to coincide with the norm of d_f^S in L_p(\mu). This generalizes previously obtained formulae---provided in [LC], 2.10 and in [L], Theorem 5---in which the distance to L_p(\mu,S) is computed in the special case of a vector subspace S of X. |
| Handle: | http://hdl.handle.net/11567/210095 |
| Appare nelle tipologie: | 01.01 - Articolo su rivista |
File in questo prodotto:
Copyright © 2021