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Note on bi-Lipschitz embeddings into normed spaces

Jiří Matoušek (1992)

Commentationes Mathematicae Universitatis Carolinae

Let ( X , d ) , ( Y , ρ ) be metric spaces and f : X Y an injective mapping. We put f Lip = sup { ρ ( f ( x ) , f ( y ) ) / d ( x , y ) ; x , y X , x y } , and dist ( f ) = f Lip . f - 1 Lip (the distortion of the mapping f ). We investigate the minimum dimension N such that every n -point metric space can be embedded into the space N with a prescribed distortion D . We obtain that this is possible for N C ( log n ) 2 n 3 / D , where C is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into p N are obtained by a similar method.

Note on operational quantities and Mil'man isometry spectrum.

Manuel González, Antonio Martinón (1991)

Extracta Mathematicae

Let X and Y be infinite dimensional Banach spaces and let L(X,Y) be the class of all (linear continuous) operators acting between X and Y. Mil'man [5] introduced the isometry spectrum I(T) of T ∈ L(X,Y) in the following way:I(T) = {α ≥ 0: ∀ ε > 0, ∃M ∈ S∞(X), ∀x ∈ SM, | ||Tx|| - α | < ε}},where S∞(X) is the set of all infinite dimensional closed subspaces of X and SM := {x ∈ M: ||x|| = 1} is the unit sphere of M ∈ S∞(X). (...)

Numerical index of vector-valued function spaces

Miguel Martín, Rafael Payá (2000)

Studia Mathematica

We show that the numerical index of a c 0 -, l 1 -, or l -sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and L 1 ( μ , X ) (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.

Numerical index with respect to an operator

Mohammad Ali Ardalani (2014)

Studia Mathematica

We introduce new concepts of numerical range and numerical radius of one operator with respect to another one, which generalize in a natural way the known concepts of numerical range and numerical radius. We study basic properties of these new concepts and present some examples.

On a dual locally uniformly rotund norm on a dual Vašák space

Marián Fabian (1991)

Studia Mathematica

We transfer a renorming method of transfer, due to G. Godefroy, from weakly compactly generated Banach spaces to Vašák, i.e., weakly K-countably determined Banach spaces. Thus we obtain a new construction of a locally uniformly rotund norm on a Vašák space. A further cultivation of this method yields the new result that every dual Vašák space admits a dual locally uniformly rotund norm.

On almost everywhere differentiability of the metric projection on closed sets in l p ( n ) , 2 < p <

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

Let F be a closed subset of n and let P ( x ) denote the metric projection (closest point mapping) of x n onto F in l p -norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in n in the Euclidean case p = 2 . We consider the case 2 < p < and prove that the i th component P i ( x ) of P ( x ) is differentiable a.e. if P i ( x ) x i and satisfies Hölder condition of order 1 / ( p - 1 ) if P i ( x ) = x i .

On Asplund functions

Wee-Kee Tang (1999)

Commentationes Mathematicae Universitatis Carolinae

A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

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