Gilles Godefroy (2020)
Commentationes Mathematicae Universitatis Carolinae
Similarity:
We show the existence of Lipschitz approximable separable spaces which fail Grothendieck's approximation property. This follows from the observation that any separable space with the metric compact approximation property is Lipschitz approximable. Some related results are spelled out.
J. A. Johnson (1979)
Colloquium Mathematicae
Similarity:
Rafał Górak (2010)
Bulletin of the Polish Academy of Sciences. Mathematics
Similarity:
We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Fréchet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gâteaux differentiable.
S. Heinrich, P. Mankiewicz (1982)
Studia Mathematica
Similarity:
Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky (2012)
Open Mathematics
Similarity:
We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
J. Grzybowski, D. Pallaschke, R. Urbański (2007)
Control and Cybernetics
Similarity:
Marco Annoni, Emanuele Casini (2007)
Studia Mathematica
Similarity:
We construct a new lipschitzian retraction from the closed unit ball of the Banach space l₁ onto its boundary, with Lipschitz constant 8.
P. Mankiewicz (1983)
Fundamenta Mathematicae
Similarity:
Nigel J. Kalton (2008)
Revista Matemática Complutense
Similarity:
Yves Dutrieux (2001)
Commentationes Mathematicae Universitatis Carolinae
Similarity:
We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets and such that is a Lipschitz-quotient of (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.
Pedro Levit Kaufmann (2015)
Studia Mathematica
Similarity:
We show that, given a Banach space X, the Lipschitz-free space over X, denoted by ℱ(X), is isomorphic to . Some applications are presented, including a nonlinear version of Pełczyński’s decomposition method for Lipschitz-free spaces and the identification up to isomorphism between ℱ(ℝⁿ) and the Lipschitz-free space over any compact metric space which is locally bi-Lipschitz embeddable into ℝⁿ and which contains a subset that is Lipschitz equivalent to the unit ball of ℝⁿ. We also show...
Nigel J. Kalton (2004)
Collectanea Mathematica
Similarity:
We study the structure of Lipschitz and Hölder-type spaces and their preduals on general metric spaces, and give applications to the uniform structure of Banach spaces. In particular we resolve a problem of Weaver who asks wether if M is a compact metric space and 0 < α < 1, it is always true the space of Hölder continuous functions of class α is isomorphic to l. We show that, on the contrary, if M is a compact convex subset of a Hilbert space this isomorphism holds if...