Normal Functionals on Lipschitz Spaces are Weak* Continuous
Autoři
Aliaga, R.J.; Pernecká, E.
Rok
2022
Publikováno
Journal of the Institute of Mathematics of Jussieu. 2022, 21(6), 2093-2102. ISSN 1474-7480.
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Anotace
Let Lip_0(M) be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in Lip_0(M)* is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of Lip_0(M) can be partially extended to Lip_0(M)*.
Topological groups with invariant linear spans
Autoři
Rok
2022
Publikováno
Revista Matemática Complutense. 2022, 35(1), 219-226. ISSN 1139-1138.
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Anotace
Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let Z(A) be the direct sum of |A|-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group Z(A) has invariant linear span. This answers a question from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. We prove that given a non-discrete sequential space X, the free abelian topological group A(X) over X is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.
Supports and extreme points in Lipschitz-free spaces
Autoři
Aliaga, R.; Pernecká, E.
Rok
2020
Publikováno
REVISTA MATEMATICA IBEROAMERICANA. 2020, 36(7), 2073-2089. ISSN 0213-2230.
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Článek
DOI
Pracoviště
Anotace
For a complete metric space M, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space FM are precisely the elementary molecules (δ(p)−δ(q))/d(p,q) defined by pairs of points p,q in M such that the triangle inequality d(p,q)
Supports in Lipschitz-free spaces and applications to extremal structure
Autoři
Aliaga, R.J.; Pernecká, E.; Petitjean, C.; Procházka, A.
Rok
2020
Publikováno
Journal of Mathematical Analysis and Applications. 2020, 489(1), ISSN 0022-247X.
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Anotace
We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space M is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space F(M). We then use this concept to study the extremal structure of F(M). We prove in particular that (δ(x) −δ(y))/d(x, y) is an exposed point of the unit ball of F(M) whenever the metric segment [x, y] is trivial, and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself. We also characterize the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of M.
Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete
Autoři
Kochanek, T.; Pernecká, E.
Rok
2018
Publikováno
Bulletin of the London Mathematical Society. 2018, 50(4), 680-696. ISSN 0024-6093.
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Článek
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Anotace
Let M be a compact subset of a superreflexive Banach space. We prove that the Lipschitz-free space F(M), the predual of the Banach space of Lipschitz functions on M, has Pełczyński's property (V*). As a consequence, F(M) is weakly sequentially complete.