Wyniki wyszukiwania

1

Random ε-nets and embeddings in $ℓ^{N}_{∞}$

100%

Gordon Y., Litvak A., Pajor A., Tomczak-Jaegermann N.

Studia Mathematica

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2007

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tom 178

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nr 1

91-98

EN

We show that, given an n-dimensional normed space X, a sequence of $N = (8/ε)^{2n}$ independent random vectors $(X_{i})_{i=1}^{N}$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ: ℝ → ℝ^{N}$ defined by $Γx = (⟨x,X_{i}⟩)_{i=1}^{N}$ embeds X in $ℓ^{N}_{∞}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{∞}^{N}$ with asymptotically best possible relation between N, n, and ε.

2

On norm closed ideals in $L(ℓ_{p},ℓ_{q})$

100%

Sari B., Schlumprecht T., Tomczak-Jaegermann N., Troitsky V.

Studia Mathematica

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2007

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tom 179

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nr 3

239-262

EN

It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L(ℓ_{p}⊕ ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L(ℓ_{p},ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L(ℓ_{p},ℓ_{q})$, including one that has not been studied before. The proofs use various methods from Banach space theory.

3

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Problems on Banach spaces

74%

Tomczak-Jaegermann N.

Colloquium Mathematicum

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1981

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tom 45

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nr 1

45-47

Ograniczanie wyników

2 Studia Mathematica

1 Colloquium Mathematicum

3 Tomczak-Jaegermann N.

1 Gordon Y.

1 Litvak A.

1 Pajor A.

1 Sari B.

1 Schlumprecht T.

1 Troitsky V.

2 2007

1 1981

Random ε-nets and embeddings in $ℓ^{N}_{∞}$

On norm closed ideals in $L(ℓ_{p},ℓ_{q})$

Problems on Banach spaces