1,720,995 research outputs found
Increasing sequences of sets and preservation of properties
Full text linkWe deal with increasing sequences of sets: by considering sequences of sets in a given class, we study when the closure of the union belongs to the same class. We consider here several classes of bounded, closed convex sets
Farthest and nearest points in balls of subspaces
No full textThe study of metric properties of the unit ball (sphere) BV (SV ) of a proper subspace
V of a Banach space X has been developed in the last decade. In this paper we give some new
results on nearest and farthest points in BV (SV ) to a point x ∈ X\V; in particular we show:
a necessary condition for a point to be critical for a distance function, a localization property
for nearest and farthest points which leads to a new characterization of Hilbert spaces among
Banach or Banach smooth spaces, detailed examples describing the phenomemon of non
uniqueness for farthest points
Parameters in Banach spaces and orthogonality
No full textIn Banach spaces, plenty of parameters have been considered: they are often dened by using
pairs of vectors. Rarely they are dened by considering pairs of vectors which are orthogonal in
the sense of Birkho and James; in that case the study is often not easy. In fact, it can be dicult
to identify pairs of orthogonal vectors; so to calculate the value of these parameters, to compare
them with the other parameters, to see if they have some stability with respect to changes of the
norm. In this paper we shall do this for a couple of new parameters.
Keywords: Orthogona
Isosceles constant in Banach spaces
No full textThe rectangular constant in Banach spaces was introduced in a paper by N. Gastinel and J.L. Joly in 1970 and has also received attention recently. To define such a constant, the notion of orthogonality, according to Birkhoff and James, is used. Here, we introduce and study a similar constant, but based on isosceles orthogonality. We indicate several properties of the new constant as a characterization of Hilbert spaces and of uniformly non-square spaces. These characterizations are similar to those proven for the old constant. Some illustrative examples complete the paper. (c) 2024 Elsevier Inc. All rights reserved
Measurements of differences between orthogonality types
No full textThe notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful. As known, when moving to normed spaces, we have many possibilities to extend this notion in a way that the new notion of orthogonality reduces to the usual one when the norm is derived from an inner product. One of these possibilities was exploited by Roberts' orthogonality. But, in a sense, it is too restrictive. Here we consider Birkhoff orthogonality and isosceles orthogonality, which are the most used notions of orthogonality, and study how "far" they are from orthogonality in the sense of Roberts. Related measurements of differences between pairs of orthogonality types are also studied. (C) 2012 Elsevier Inc. All rights reserved.Mathematics, AppliedMathematicsSCI(E)0ARTICLE1285-29139
Rascal finite sets
No full textWe collect examples of sets, in Bnach spaces, with some strange propertie
Bad normed spaces, convexity properties, separated sets
No full textWe discuss some parameters in Banach spaces and their relations with the geometry of the spac
Equilateral, diametral, centered sets and subsets of spheres
No full textWe study properties implying that a set in a Banach space lies on a sphere
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