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Linearization and compactness

Jesús Ángel Jaramillo, Ángeles Prieto, Ignacio Zalduendo (2009)

Studia Mathematica

This paper is devoted to several questions concerning linearizations of function spaces. We first consider the relation between linearizations of a given space when it is viewed as a function space over different domains. Then we study the problem of characterizing when a Banach function space admits a Banach linearization in a natural way. Finally, we consider the relevance of compactness properties in linearizations, more precisely, the relation between different compactness properties of a mapping,...

Linearization of isometric embedding on Banach spaces

Yu Zhou, Zihou Zhang, Chunyan Liu (2015)

Studia Mathematica

Let X,Y be Banach spaces, f: X → Y be an isometry with f(0) = 0, and T : s p a n ¯ ( f ( X ) ) → X be the Figiel operator with T ∘ f = I d X and ||T|| = 1. We present a sufficient and necessary condition for the Figiel operator T to admit a linear isometric right inverse. We also prove that such a right inverse exists when s p a n ¯ ( f ( X ) ) is weakly nearly strictly convex.

Linearly rigid metric spaces and the embedding problem

J. Melleray, F. V. Petrov, A. M. Vershik (2008)

Fundamenta Mathematicae

We consider the problem of isometric embedding of metric spaces into Banach spaces, and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a...

Lineární posloupnosti

Miroslav Laitoch (1968)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica-Physica-Chemica

Lions-Peetre reiteration formulas for triples and their applications

Irina Asekritova, Natan Krugljak, Lech Maligranda, Lyudmila Nikolova, Lars-Erik Persson (2001)

Studia Mathematica

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted L p -spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability...

Lipschitz and uniform embeddings into ℓ ∞

N. J. Kalton (2011)

Fundamenta Mathematicae

We show that there is no uniformly continuous selection of the quotient map Q : ℓ ∞ → ℓ ∞ / c ₀ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from B X * * onto B X .

Lipschitz approximable Banach spaces

Gilles Godefroy (2020)

Commentationes Mathematicae Universitatis Carolinae

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.

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

We study continuity envelopes of function spaces B p , q s ( ℝ ⁿ , w ) and F p , q s ( ℝ ⁿ , w ) where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz M , definita su un sottoinsieme di uno spazio di Hilbert H a valori compatti e convessi in ℝ n , può essere estesa su tutto H ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz M .

Lipschitz-free Banach spaces

G. Godefroy, N. J. Kalton (2003)

Studia Mathematica

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y, then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation...

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