Notions of denseness

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Notions of denseness

The notion of a completely saturated packing [1] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent dense packings and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, ...

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Denseness for norm attaining operator-valued functions

In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss’ famous result on the denseness of norm attaining operators. Specifically, we show given any A ∈ L(H) there is a sequence of rank-1 operators Kn such that A+Kn is norm attaining for each n and Kn converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operato...

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On the denseness of Jacobi polynomials

Let X represent either a space C[−1,1] or L α,β(w), 1 ≤ p < ∞, of functions on [−1,1]. It is well known that X are Banach spaces under the sup and the p-norms, respectively. We prove that there exist the best possible normalized Banach subspaces Xα,β of X such that the system of Jacobi polynomials is densely spread on these, or, in other words, each f ∈ Xα,β can be represented by a linear combi...

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Denseness of Numerical Radius Attaining Holomorphic Functions

LetX be a complex Banach space andX∗ its dual space. We consider the topological subspace Π X { x, x∗ : x∗ x 1 ‖x‖ ‖x∗‖} of the product space BX ×BX∗ , equipped with norm and weak-∗ topology on the unit ball BX ofX and its dual unit ball BX∗ , respectively. It is easy to see that Π X is a closed subspace of BX × BX∗ . For two complex Banach spaces X and Y , denote by Cb BX : Y the Banach space ...

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ژورنال

عنوان ژورنال: Geometry & Topology

سال: 2000

ISSN: 1364-0380,1465-3060

DOI: 10.2140/gt.2000.4.277