Banach

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[Banach] Abstract of a paper by S. Artstein, V. Milman, S. J. Szarek, and N. Tomczak-Jaegermann
by Dale Alspach 15 Jul '04

15 Jul '04

This is an announcement for the paper "On convexified packing and entropy duality" by S. Artstein, V. Milman, S. J. Szarek, and N. Tomczak-Jaegermann. Abstract: For a compact operator acting between two Banach spaces, a 1972 duality conjecture due to Pietsch asserts that its entropy numbers and those of its adjoint are equivalent. This is equivalent to a dimension-free inequality relating covering (or packing) numbers for convex bodies to those of their polars. The duality conjecture has been recently proved (see math.FA/0407236) in the central case when one of the Banach spaces is Hilbertian, which - in the geometric setting - corresponds to a duality result for symmetric convex bodies in Euclidean spaces. In the present paper we define a new notion of "convexified packing," show a duality theorem for that notion, and use it to prove the duality conjecture under much milder conditions on the spaces involved (namely, that one of them is K-convex). Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B10; 46B07; 46B50; 47A05; 52C17; 51F99 Remarks: 6 p., LATEX The source file(s), ConvPackShort5.tex: 21620 bytes, is(are) stored in gzipped form as 0407238.gz with size 8kb. The corresponding postcript file has gzipped size 43kb. Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407238 or http://arXiv.org/abs/math.FA/0407238 or by email in unzipped form by transmitting an empty message with subject line uget 0407238 or in gzipped form by using subject line get 0407238 to: math(a)arXiv.org.

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Abstract of a paper by S. Artstein, V. Milman, and S. J. Szarek
by Dale Alspach 15 Jul '04

15 Jul '04

This is an announcement for the paper "Duality of metric entropy" by S. Artstein, V. Milman, and S. J. Szarek. Abstract: For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in $R^n$. We prove that the two functions of t, N(K,tD) and N(D, tK^o), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K in $R^n$ and over positive integers n. In particular, this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B10; 47A05; 52C17; 51F99 Remarks: 17 p., LATEX The source file(s), ArtMilSzaAoM.tex: 40692 bytes, is(are) stored in gzipped form as 0407236.gz with size 14kb. The corresponding postcript file has gzipped size 68kb. Submitted from: szarek(a)cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407236 or http://arXiv.org/abs/math.FA/0407236 or by email in unzipped form by transmitting an empty message with subject line uget 0407236 or in gzipped form by using subject line get 0407236 to: math(a)arXiv.org.

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Abstract of a paper by Stanislaw Szarek
by Dale Alspach 14 Jul '04

14 Jul '04

This is an announcement for the paper "The volume of separable states is super-doubly-exponentially small" by Stanislaw Szarek. Abstract: In this note we give sharp estimates on the volume of the set of separable states on N qubits. In particular, the magnitude of the "effective radius" of that set in the sense of volume is determined up to a factor which is a (small) power of N, and thus precisely on the scale of powers of its dimension. Additionally, one of the appendices contains sharp estimates (by known methods) for the expected values of norms of the GUE random matrices. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces. Archive classification: Quantum Physics; Functional Analysis Remarks: 20 p., LATEX; an expanded version of the original submission: more background material from convexity and geometry of Banach spaces, more exhaustive bibliography and improved quality of references to the bibliography The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/quant-ph/0310061 or http://arXiv.org/abs/quant-ph/0310061 or by email in unzipped form by transmitting an empty message with subject line uget /0310061 or in gzipped form by using subject line get /0310061 to: math(a)arXiv.org.

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Abstract of a paper by Jeremy J. Becnel
by Dale Alspach 13 Jul '04

13 Jul '04

This is an announcement for the paper "About countably-normed spaces" by Jeremy J. Becnel. Abstract: Here we present an overview of countably normed spaces. In particular, we discuss the main topologies---weak, strong, inductive, and Mackey---placed on the dual of a countably normed spaces and discuss the sigma fields generated by these topologies. In particlar, we show that the strong, inductive, and Mackey topologies are equivalent under reasonable conditions. Also we show that all four topologies induce the same Borel field under certain conditions. The purpose in mind is to provide the background material for many of the results used in White Noise Analysis. Archive classification: Functional Analysis Mathematics Subject Classification: 46A11 Remarks: 23 pages, 0 figures, Background material for White Noise Analysis The source file(s), NuclearSpace.bbl: 1198 bytes, NuclearSpace.tex: 1472 bytes, borel.tex: 5271 bytes, cns.tex: 16479 bytes, compare.tex: 6600 bytes, conclusion.tex: 4430 bytes, inductive.tex: 6567 bytes, nuclear.sty: 4578 bytes, strong.tex: 17400 bytes, tvs.tex: 14418 bytes, weak.tex: 3536 bytes, is(are) stored in gzipped form as 0407200.tar.gz with size 23kb. The corresponding postcript file has gzipped size 103kb. Submitted from: beck(a)math.lsu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407200 or http://arXiv.org/abs/math.FA/0407200 or by email in unzipped form by transmitting an empty message with subject line uget 0407200 or in gzipped form by using subject line get 0407200 to: math(a)arXiv.org.

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Abstract of a paper by Christian Rosendal
by Dale Alspach 08 Jul '04

08 Jul '04

This is an announcement for the paper "Incomparable, non isomorphic and minimal Banach spaces" by Christian Rosendal. Abstract: A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes and has an isomorphically homogeneous subsequence. Archive classification: Functional Analysis; Logic The source file(s), ArchiveIncomparable.tex: 57150 bytes, is(are) stored in gzipped form as 0407111.gz with size 19kb. The corresponding postcript file has gzipped size 81kb. Submitted from: rosendal(a)ccr.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407111 or http://arXiv.org/abs/math.FA/0407111 or by email in unzipped form by transmitting an empty message with subject line uget 0407111 or in gzipped form by using subject line get 0407111 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi and Eloi Medina Galego
by Dale Alspach 25 Jun '04

25 Jun '04

This is an announcement for the paper "Some results about the Schroeder-Bernstein Property for separable Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of V. Ferenczi and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder-Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition. Archive classification: Functional Analysis Mathematics Subject Classification: 46B03, 46B20 Remarks: 25 pages The source file(s), ferenczigalegoSB.tex: 74499 bytes, is(are) stored in gzipped form as 0406479.gz with size 22kb. The corresponding postcript file has gzipped size 87kb. Submitted from: eloi(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0406479 or http://arXiv.org/abs/math.FA/0406479 or by email in unzipped form by transmitting an empty message with subject line uget 0406479 or in gzipped form by using subject line get 0406479 to: math(a)arXiv.org.

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Abstract of a paper by Valentin Ferenczi and Eloi Medina Galego
by Dale Alspach 25 Jun '04

25 Jun '04

This is an announcement for the paper "Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that the relation $E_{K_{\sigma}}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or $\ell_p, 1 \leq p <2$. We show that the relation $E_{K_{\sigma}} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 03E15; 46B03 Remarks: 22 pages; 2 figures The source file(s), sjm16.tex: 74499 bytes, is(are) stored in gzipped form as 0406477.gz with size 22kb. The corresponding postcript file has gzipped size 86kb. Submitted from: eloi(a)ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0406477 or http://arXiv.org/abs/math.FA/0406477 or by email in unzipped form by transmitting an empty message with subject line uget 0406477 or in gzipped form by using subject line get 0406477 to: math(a)arXiv.org.

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Abstract of a paper by Artem Zvavitch
by Dale Alspach 22 Jun '04

22 Jun '04

This is an announcement for the paper "The Busemann-Petty problem for arbitrary measures" by Artem Zvavitch. Abstract: The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in $\R^n$ and solve a version of generalized Busemann-Petty problem for sections by k-dimensional subspaces. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52A15, 52A21, 52A38 The source file(s), GBP_Zvavitch.tex: 44254 bytes, is(are) stored in gzipped form as 0406406.gz with size 12kb. The corresponding postcript file has gzipped size 65kb. Submitted from: zvavitch(a)math.kent.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0406406 or http://arXiv.org/abs/math.MG/0406406 or by email in unzipped form by transmitting an empty message with subject line uget 0406406 or in gzipped form by using subject line get 0406406 to: math(a)arXiv.org.

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Abstract of a paper by Peter A. Loeb and Erik Talvila
by Dale Alspach 21 Jun '04

21 Jun '04

This is an announcement for the paper "Lusin's Theorem and Bochner integration" by Peter A. Loeb and Erik Talvila. Abstract: It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned $\varepsilon$ of the integral, with the sum for the local errors also less than $\varepsilon$. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion. Archive classification: Classical Analysis and ODEs; Functional Analysis Mathematics Subject Classification: 28A20, 28B05; 26A39 Remarks: To appear in Scientiae Mathematicae Japonicae The source file(s), bochnerbox.tex: 34366 bytes, is(are) stored in gzipped form as 0406370.gz with size 11kb. The corresponding postcript file has gzipped size 52kb. Submitted from: etalvila(a)math.ualberta.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.CA/0406370 or http://arXiv.org/abs/math.CA/0406370 or by email in unzipped form by transmitting an empty message with subject line uget 0406370 or in gzipped form by using subject line get 0406370 to: math(a)arXiv.org.

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Abstract of a paper by Konrad J. Swanepoel
by Dale Alspach 16 Jun '04

16 Jun '04

This is an announcement for the paper "Equilateral sets in finite-dimensional normed spaces" by Konrad J. Swanepoel. Abstract: This is an expository paper on the largest size of equilateral sets in finite-dimensional normed spaces. Archive classification: Metric Geometry; Functional Analysis Mathematics Subject Classification: 52A21 (Primary) 46B20, 52C17 (Secondary) Remarks: 30 pages The source file(s), equilateral.tex: 94432 bytes, is(are) stored in gzipped form as 0406264.gz with size 29kb. The corresponding postcript file has gzipped size 128kb. Submitted from: swanekj(a)unisa.ac.za The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.MG/0406264 or http://arXiv.org/abs/math.MG/0406264 or by email in unzipped form by transmitting an empty message with subject line uget 0406264 or in gzipped form by using subject line get 0406264 to: math(a)arXiv.org.

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