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Uncountable sets of unit vectors that are separated by more than 1

Tomasz Kania, Tomasz Kochanek (2016)

Studia Mathematica

Let X be a Banach space. We study the circumstances under which there exists an uncountable set 𝓐 ⊂ X of unit vectors such that ||x-y|| > 1 for any distinct x,y ∈ 𝓐. We prove that such a set exists if X is quasi-reflexive and non-separable; if X is additionally super-reflexive then one can have ||x-y|| ≥ slant 1 + ε for some ε > 0 that depends only on X. If K is a non-metrisable compact, Hausdorff space, then the unit sphere of X = C(K) also contains such a subset; if moreover K is perfectly...

Une nouvelle classe d'espaces de Banach vérifiant le théorème de Grothendieck

Gilles Pisier (1978)

Annales de l'institut Fourier

Soit W un espace ℒ 1 et soit R un sous-espace réflexif de dimension infinie de W . Nous montrons que le quotient W / R vérifie le théorème de Grothendieck, c’est-à-dire que tout opérateur de W / R dans un espace de Hilbert est 1-sommant; par ailleurs, W / R n’est pas un espace ℒ 1 . Cela permet de répondre négativement à une question de Lindenstrauss-Pełczyński ainsi qu’à une question similaire de Grothendieck.

Uniform bounds for the bilinear Hilbert transforms (II).

Xiaochun Li (2006)

Revista Matemática Iberoamericana

We continue the investigation initiated in [Grafakos, L. and Li, X.: Uniform bounds for the bilinear Hilbert transforms (I). Ann. of Math. (2)159 (2004), 889-933] of uniform Lp bounds for the family of bilinear Hilbert transformsHα,β(f,g)(x) = p.v. ∫R f(x - αt) g (x - βt) dt/t.

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