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Abstract of a paper by Michael Cwikel and Eliahu Levy
by alspach＠fourier.math.okstate.edu 04 Nov '08

04 Nov '08

This is an announcement for the paper "Estimates for covering numbers in Schauder's theorem about adjoints of compact operators" by Michael Cwikel and Eliahu Levy. Abstract: Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y' --> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y' respectively. We obtain apparently new estimates for the covering numbers of the set S(B(Y')). These are expressed in terms of the covering numbers of T(B(X)), or, more generally, in terms of the covering numbers of a "significant" subset of T(B(X)). The latter more general estimates are best possible. These estimates follow from our new quantitative version of an abstract compactness result which generalizes classical theorems of Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of a suitable "significant" subset of S(B(Y')). Archive classification: math.FA Mathematics Subject Classification: Primary 46B06. Secondary 46B10, 46B50, 05B40, 52C17, 52C15. Remarks: 14 pages. At any given time our most recent version of this paper will be either at http://www.math.technion.ac.il/~mcwikel/compact/QuantitativeSchauder.pdf or http://arxiv.org/abs/0810.4240 The source file(s), 8QuantitativeSchauder.tex: 51761 bytes, is(are) stored in gzipped form as 0810.4240.gz with size 15kb. The corresponding postcript file has gzipped size 105kb. Submitted from: mcwikel(a)math.technion.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.4240 or http://arXiv.org/abs/0810.4240 or by email in unzipped form by transmitting an empty message with subject line uget 0810.4240 or in gzipped form by using subject line get 0810.4240 to: math(a)arXiv.org.

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Abstract of a paper by R. Fry
by alspach＠fourier.math.okstate.edu 04 Nov '08

04 Nov '08

This is an announcement for the paper "Approximation by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces" by R. Fry. Abstract: It is shown that on weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded, real-valued functions by Lipschitz, C^{p} smooth functions. This provides a `Lipschitz version' of the classical approximation results of Godefroy, Troyanski, Whitfield and Zizler. Archive classification: math.FA Mathematics Subject Classification: 46B20 Citation: Journal of Functional Analysis, Volume 252, Issue 1, 1 November The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.3901 or http://arXiv.org/abs/0810.3901 or by email in unzipped form by transmitting an empty message with subject line uget 0810.3901 or in gzipped form by using subject line get 0810.3901 to: math(a)arXiv.org.

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Abstract of a paper by Dusan Repovs and Pavel V. Semenov
by alspach＠fourier.math.okstate.edu 04 Nov '08

04 Nov '08

This is an announcement for the paper "On continuous choice of retractions onto nonconvex subsets" by Dusan Repovs and Pavel V. Semenov. Abstract: For a Banach space $B$ and for a class $\A$ of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements $A \in \A$ can be chosen to depend continuously on $A$, whenever nonconvexity of each $A \in \A$ is less than $\f{1}{2}$. The key geometric argument is that the set of all uniform retractions onto an $\a-$paraconvex set (in the spirit of E. Michael) is $\frac{\a}{1-\a}-$paraconvex subset in the space of continuous mappings of $B$ into itself. For a Hilbert space $H$ the estimate $\frac{\a}{1-\a}$ can be improved to $\frac{\a (1+\a^{2})}{1-\a^{2}}$ and the constant $\f{1}{2}$ can be reduced to the root of the equation $\a+ \a^{2}+a^{3}=1$. Archive classification: math.GN math.FA Mathematics Subject Classification: 54C60; 54C65; 41A65; 54C55; 54C20 The source file(s), VerzijaZaArhiv.tex: 38914 bytes, is(are) stored in gzipped form as 0810.3895.gz with size 12kb. The corresponding postcript file has gzipped size 89kb. Submitted from: dusan.repovs(a)guest.arnes.si The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.3895 or http://arXiv.org/abs/0810.3895 or by email in unzipped form by transmitting an empty message with subject line uget 0810.3895 or in gzipped form by using subject line get 0810.3895 to: math(a)arXiv.org.

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Abstract of a paper by R. Fry
by alspach＠fourier.math.okstate.edu 28 Oct '08

28 Oct '08

This is an announcement for the paper "Corrigendum to Approximation by C^{p}-smooth, Lipschitz functions on Banach spaces" [J. Math. Anal. Appl., 315 (2006), 599-605]" by R. Fry. Abstract: In this erratum, we recover the results from an earlier paper of the author's which contained a gap. Specifically, we prove that if X is a Banach space with an unconditional basis and admits a C^{p}-smooth, Lipschitz bump function, and Y is a convex subset of X, then any uniformly continuous function f: Y->R can be uniformly approximated by Lipschitz, C^{p}-smooth functions K:X->R. Also, if Z is any Banach space and f:X->Z is L-Lipschitz, then the approximates K:X->Z can be chosen CL-Lipschitz and C^{p}-smooth, for some constant C depending only on X. Archive classification: math.FA Mathematics Subject Classification: 46B20 Citation: Journal of Mathematical Analysis and Applications, Volume 348, The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.3881 or http://arXiv.org/abs/0810.3881 or by email in unzipped form by transmitting an empty message with subject line uget 0810.3881 or in gzipped form by using subject line get 0810.3881 to: math(a)arXiv.org.

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Abstract of a paper by Siu-Ah Ng
by alspach＠fourier.math.okstate.edu 28 Oct '08

28 Oct '08

This is an announcement for the paper "Bidual as a weak nonstandard hull" by Siu-Ah Ng. Abstract: We construct the weak nonstandard hull of a normed linear space X from *X (the nonstandard extension of X) using the weak topology on X. The bidual (i.e. the second dual) X'' is shown to be isometrically isomorphic to the weak nonstandard hull of X. Examples and applications to C*-algebras are given, including a simple proof of the Sherman-Takeda Theorem. As a consequence, the weak nonstandard hull of a C*-algebra is always a von Neumann algebra. Moreover a natural representation of the Arens product is given. Archive classification: math.FA math.LO math.OA Mathematics Subject Classification: 46L05, 03H05, 26E3,5 46S20 Remarks: 14 pages The source file(s), bidual.tex: 38768 bytes, is(are) stored in gzipped form as 0810.3090.gz with size 11kb. The corresponding postcript file has gzipped size 87kb. Submitted from: ngs(a)ukzn.ac.za The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.3090 or http://arXiv.org/abs/0810.3090 or by email in unzipped form by transmitting an empty message with subject line uget 0810.3090 or in gzipped form by using subject line get 0810.3090 to: math(a)arXiv.org.

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Abstract of a paper by Javier Guachalla H
by alspach＠fourier.math.okstate.edu 28 Oct '08

28 Oct '08

This is an announcement for the paper "$L^1$ is complemented in $L^{\infty *}$" by Javier Guachalla H. Abstract: We show $L^1$ is complemented in the dual space $L^{\infty *}$ Archive classification: math.FA The source file(s), l1cmplm.TEX: 2401 bytes, is(are) stored in gzipped form as 0810.2354.gz with size 1kb. The corresponding postcript file has gzipped size 29kb. Submitted from: jguachallah(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.2354 or http://arXiv.org/abs/0810.2354 or by email in unzipped form by transmitting an empty message with subject line uget 0810.2354 or in gzipped form by using subject line get 0810.2354 to: math(a)arXiv.org.

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Abstract of a paper by Libor Vesely and Ludek Zajicek
by alspach＠fourier.math.okstate.edu 28 Oct '08

28 Oct '08

This is an announcement for the paper "On extensions of d.c. functions and convex functions" by Libor Vesely and Ludek Zajicek. Abstract: We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems, concerning extendability of continuous convex functions from a closed subspace of a normed linear space, complement recent results of J.Borwein, V.Montesinos and J.Vanderwerff. Archive classification: math.FA math.GM Mathematics Subject Classification: 52A41; 26B25; 46B99 Remarks: 16 pages The source file(s), RozsirDCfinal.tex: 48466 bytes, is(are) stored in gzipped form as 0810.1433.gz with size 15kb. The corresponding postcript file has gzipped size 110kb. Submitted from: Libor.Vesely(a)mat.unimi.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.1433 or http://arXiv.org/abs/0810.1433 or by email in unzipped form by transmitting an empty message with subject line uget 0810.1433 or in gzipped form by using subject line get 0810.1433 to: math(a)arXiv.org.

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Abstract of a paper by Hadi Haghshenas
by alspach＠fourier.math.okstate.edu 28 Oct '08

28 Oct '08

This is an announcement for the paper "Convexity of Chebyshev sets in Hilbert spaces" by Hadi Haghshenas. Abstract: The aim of this paper is state of conditions that ensure the convexity of a Chebyshev sets in Hilbert spaces . Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 4 Pages The source file(s), CONVEXITYOFCEBYSEVSETSINHILBERTSPACES.tex: 8784 bytes, is(are) stored in gzipped form as 0810.0772.gz with size 3kb. The corresponding postcript file has gzipped size 36kb. Submitted from: h_haghshenas60(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.0772 or http://arXiv.org/abs/0810.0772 or by email in unzipped form by transmitting an empty message with subject line uget 0810.0772 or in gzipped form by using subject line get 0810.0772 to: math(a)arXiv.org.

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Abstract of a paper by A. Assadi, HADI Haghshenas and H. Hosseini Guive
by alspach＠fourier.math.okstate.edu 28 Oct '08

28 Oct '08

This is an announcement for the paper "A review on some geometric results of the Smulian's theorem on Frechet differentiability of norms" by A. Assadi,HADI Haghshenas and H. Hosseini Guive. Abstract: In this paper, we prove the Smulian s theorem on Frechet differentiability of norm,and present some of its geometric results concerning the Gateaux and Frechet differentiability of norm and properties of the allied space and its dual such as reflexivity and strict convexity. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 4 Pages The source file(s), AREVIEWONSOMEGEOMETRICRESULTSOFTHESMULIANSTHEOREMONFRECHETDIFFERENTIABILITYOFNORMS.tex: 10646 bytes, is(are) stored in gzipped form as 0810.0773.gz with size 4kb. The corresponding postcript file has gzipped size 42kb. Submitted from: h_haghshenas60(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.0773 or http://arXiv.org/abs/0810.0773 or by email in unzipped form by transmitting an empty message with subject line uget 0810.0773 or in gzipped form by using subject line get 0810.0773 to: math(a)arXiv.org.

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Abstract of a paper by Assadi, Hadi Haghshenas, and Hosseini Guive
by alspach＠fourier.math.okstate.edu 07 Oct '08

07 Oct '08

This is an announcement for the paper "Convexity of Chebyshev sets through differentiability of distance function" by Assadi, Hadi Haghshenas, and Hosseini Guive. Abstract: The aim of this paper is to present some equivalent conditions that ensure the convexity of a Chebyshev set. To do so, we use Gateaux differentiability of the distance function Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 4 pages The source file(s), CONVEXITYOFCEBYSEVSETSTHROUGHDIFFERENTIABILITYOFDISTANCEFUNCTION.tex: 10884 bytes, is(are) stored in gzipped form as 0810.0587.gz with size 4kb. The corresponding postcript file has gzipped size 41kb. Submitted from: h_haghshenas60(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0810.0587 or http://arXiv.org/abs/0810.0587 or by email in unzipped form by transmitting an empty message with subject line uget 0810.0587 or in gzipped form by using subject line get 0810.0587 to: math(a)arXiv.org.

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