Carlo Franchetti

Projections onto hyperplanes in Banach spaces

On presente des resultats relies aux projections sur un hyperplan et on donne des relations entre les normes des projections, la forme de la boule unite et les proprietes metriques des hyperplans

The embedding of proximinal sets

Abstract A subset M of a normed linear space X is said to be proximinal if inf m ϵ M ∥ x − m ∥ is attained, for each x ϵ X . If X is embedded in another normed space, Z , a proximinal subset of X may or may not be proximinal in Z . Certain practical problems in multivariate approximation lead us to examine the case when X = C ( S ) and Z = C ( S × T ), where S and T are compact Hausdorff spaces. We characterize the proximinal subspaces of C ( S ) which are proximinal in C ( S × T ) for every T . In another section, a generalization of Mazurs Proximinality Theorem is given. This generalization gives a condition under which a subspace of functions ν ° f is proximinal, when f is held fixed and ν ranges over a proximinal subspace.

The retraction constant in some Banach spaces

Abstract We give new sharper estimations for the retraction constant in some Banach spaces.

Estimates for the Jung constant in Banach lattices

Sharp lower estimates for the Jung constantJ(E) in Banach latticesE satisfying an upperp-estimate and a lowerq-estimate are given. Moreover, the minimal value ofJ(E) with respect to equivalent renormings ofE is calculated inE=Lp,q for finitep andq, as well as in more general spacesE. Finally, a nontrivial estimate for the radiusrLp,∞(A) is obtained forA being a bounded sequence of disjointly supported functions inLp,∞.

Una nuova proprieta' geometrica della sfera unitaria di uno spazio normato

SuntoFra gli spazi di Banach reali vengono definiti iT-spazi. Tale classe include gli spazi finito-dimensionali e quelli uniformemente lisci. Riguardo alla proprietàT e a proprietà affini vengono discussi alcuni spazi di Banach classici. Sono provate proprietà deiT-spazi riguardo le trasformazioni lipschitziane del disco unitario dello spazio in sè.SummaryWe define theT-spaces among the real Banach spaces. This class includes finite dimensional and uniformly smooth spaces. We discuss some classical Banach spaces with respect theT-property and related properties. We prove some properties of theT-spaces concerning Lipschitz self maps of the unit ball of the space.

Alcuni risultati sugli insiemi prossiminali negli spazi di Banach

SommarioScopo di questo lavoro è presentare alcuni recenti risultati sugli insiemi prossiminali negli spazi di Banach.Speciale attenzione è dedicata al caso dello spazio delle funzioni reali continue: in particolare si introducono delle proprietà più forti della prossiminalità che consentono di dimostrare risultati nuovi anche nel caso della semplice prossiminalità.Dato il carattere espositivo del lavoro, non sono in genere riportate le dimostrazioni dei teoremi.SummaryAim of this paper is to present some recent results on proximinal subsets of Banach spaces. The space of real continuous functions is especially considered: in particular some properties which are stronger than proximinality are introduced. By means of these properties some new results are proved also in the case of the usual proximinality.Due to the expository character of this paper the proofs of the theorems are generally omitted.