Search results for "Dual norm"

Norm estimates for operators from Hp to ℓq

2008

Abstract We give upper and lower estimates of the norm of a bounded linear operator from the Hardy space H p to l q in terms of the norm of the rows and the columns of its associated matrix in certain vector-valued sequence spaces.

A General Framework for the One Center Location Problem

1992

This paper deals with an optimization problem where the objective function F is defined on a real vector space X by F(x) = γ(w 1║x - a 1║1, ⋯, w n ║x - a n║ n ), a formula in which a 1, ⋯, a n are n given points in X, ║∙║1, ⋯, ║∙║ n n norms on X, w 1, ⋯, w n positive numbers and γ a monotone norm on ℝ n . A geometric description of the set of optimal solutions to the problem min F(x) is given, illustrated by some examples. When all norms ║∙║i are equal, and γ being successively the l 1 , l ∞ and l 2-norm, a particular study is made, which shows the peculiar role played by the l 1-norm.

Transformations by diagonal matrices in a normed space

1962

An overdetermined problem for the anisotropic capacity

2015

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in \({\mathbb {R}}^{N}\), establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm \(H_0\)).

Almost square dual Banach spaces

2020

Abstract We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on l ∞ . As a consequence we get that every dual Banach space containing c 0 has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.

Some Open Problems

2009

We have extensively considered here the use of Stone's theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known cases of existence of such a renorming property [Hay99, MOTV06]. Our approximation process is based on co-σ-continuous maps using that they have separable fibers, see Sect. 2.2. We present now some problems that remain open in this area. Some of them are classical and have been asked by different authors in conferences, papers and books. Others h…

Norm continuity and related notions for semigroups on Banach spaces

1996

We find some conditions on a c0-semigroup on a Banach space and its resolvent connected with the norm continuity of the semigroup. We use them to get characterizations of norm continuous, eventually norm continuous and eventually compact semigroups on Hilbert spaces in terms of the growth of the resolvent of their generator.