Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Volker Wilhelm Thurey
by alspach＠math.okstate.edu 13 Jul '12

13 Jul '12

This is an announcement for the paper "Angles and a classification of normed spaces" by Volker Wilhelm Thurey. Abstract: We suggest a concept of generalized `angles' in arbitrary real normed vector spaces. We give for each real number a definition of an `angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the special case of real inner product spaces. With these different angles we achieve a classification of normed spaces, and we obtain a characterization of inner product spaces. Finally we consider this construction also for a generalization of normed spaces, i.e. for spaces which may have a non-convex unit ball. Archive classification: math.FA Mathematics Subject Classification: 2010 AMS-classification: 46B20, 52A10 Remarks: 23 pages, 1 figure Submitted from: volker(a)thuerey.de The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1207.0074 or http://arXiv.org/abs/1207.0074

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Abstract of a paper by Rui Liu
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "Hilbert-Schauder frame operators" by Rui Liu. Abstract: We introduce a new concept of frame operators for Banach spaces we call a Hilbert-Schauder frame operator. This is a hybird between standard frame theory for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our results involve basic structure properties of the Hilbert-Schauder frame operator. Examples of Hilbert-Schauder frames include standard Hilbert frames and classical bases of $\ell_p$ and $L^p$-spaces with $1< p \le 2$. Finally, we give a new isomorphic characterization of Hilbert spaces. Archive classification: math.FA math.CA math.OA Mathematics Subject Classification: 46B, 47B, 47A Remarks: 9 pages, to appear in Operators and Matrices Submitted from: ruiliu(a)nankai.edu.cn The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.6146 or http://arXiv.org/abs/1206.6146

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Abstract of a paper by Ivan S. Feshchenko
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "On absolutely representing families of subspaces in Banach spaces" by Ivan S. Feshchenko. Abstract: An absolutely representing family of subspaces is a natural generalization of an absolutely representing system of subspaces and absolutely representing system (of elements). We obtain necessary and (or) sufficient conditions for a family of subspaces to be an absolutely representing family of subspaces and study properties of absolutely representing families of subspaces in Banach spaces. As an example, we study families of subspaces spanned by exponents. Archive classification: math.FA Mathematics Subject Classification: 41A58, 46B99 Remarks: 15 pages, submitted to Vladikavkaz Mathematical Journal Submitted from: ivanmath007(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.5496 or http://arXiv.org/abs/1206.5496

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Abstract of a paper by Tomasz Kania and Tomasz Kochanek
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "The ideal of weakly compactly generated operators acting on a Banach space" by Tomasz Kania and Tomasz Kochanek. Abstract: We call a bounded linear operator acting between Banach spaces weakly compactly generated ($\mathsf{WCG}$ for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of $\mathsf{WCG}$ operators, we prove that it forms a closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the $p$th long James space $\mathcal{J}_p(\omega_1)$, we show how properties of the ideal of $\mathsf{WCG}$ operators (such as being the unique maximal ideal) may be used to derive results outside ideal theory. For instance, we identify the $K_0$-group of $\mathscr{B}(\mathcal{J}_p(\omega_1))$ as the additive group of integers. Archive classification: math.FA math.OA Mathematics Subject Classification: Primary 47L10, 47L20, Secondary 46H10, 46B26 Submitted from: t.kania(a)lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.5424 or http://arXiv.org/abs/1206.5424

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Abstract of a paper by Alexander Barvinok
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "Thrifty approximations of convex bodies by polytopes" by Alexander Barvinok. Abstract: Given a convex body C in R^d we construct a polytope P in C with relatively few vertices which approximates C relatively well. In particular, we prove that if C=-C then for any 1>epsilon>0 to have P in C and C in (1+epsilon) P one can choose P having roughly epsilon^{-d/2} vertices and for P in C and C in sqrt{epsilon d} P one can choose P having roughly d^{1/epsilon} vertices. Similarly, we prove that if C in R^d is a convex body such that -C in mu C for some mu > 1 then to have P in C and C in (1+epsilon)P one can choose P having roughly (mu/epsilon)^{d/2} vertices. Archive classification: math.MG math.CO math.FA Mathematics Subject Classification: 52A20, 52A27, 52A21, 52B55 Remarks: 13 pages Submitted from: barvinok(a)umich.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.3993 or http://arXiv.org/abs/1206.3993

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Abstract of a paper by Fernando Albiac and Florent Baudier
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces $L_p$ and $\ell_{p}$ for $0<p<\infty$" by Fernando Albiac and Florent Baudier. Abstract: We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range $0<p<\infty$ from a metric viewpoint and give a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with both their ad-hoc distances and their snowflakings. Through connections with weaker forms of embeddings that lead to basic (yet fundamental) open problems, we also set the challenging goal of understanding the dissimilarities between the well-known subspace structure and the different nonlinear geometries that coexist inside $L_{p}$ and $\ell_{p}$. Archive classification: math.MG math.FA Mathematics Subject Classification: 46B80, 46A16, 46T99 Remarks: 25 pages Submitted from: florent(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.3774 or http://arXiv.org/abs/1206.3774

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Abstract of a paper by Trond A. Abrahamsen Vegard Lima, and Olav Nygaard
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "Super-ideals in Banach spaces" by Trond A. Abrahamsen Vegard Lima, and Olav Nygaard. Abstract: A natural class of ideals, super-ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observations that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces which are ideals in every superspace; we show that being a super-ideal in every superspace characterizes the class of Gurarii spaces. Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 14 pages Submitted from: veli(a)hials.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.3539 or http://arXiv.org/abs/1206.3539

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Abstract of a paper by Daniel Carando, Silvia Lassalle and Martin Mazzitelli
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "On the polynomial Lindenstrauss theorem" by Daniel Carando, Silvia Lassalle and Martin Mazzitelli. Abstract: Under certain hypotheses on the Banach space $X$, we show that the set of $N$-homogeneous polynomials from $X$ to any dual space, whose Aron-Berner extensions are norm attaining, is dense in the space of all continuous $N$-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop-Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollob\'as, of these results. Archive classification: math.FA Submitted from: mmazzite(a)dm.uba.ar The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.3218 or http://arXiv.org/abs/1206.3218

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Abstract of a paper by Dmitry V. Rutsky
by alspach＠math.okstate.edu 28 Jun '12

28 Jun '12

This is an announcement for the paper "Linear selections of superlinear set-valued maps with some applications to analysis" by Dmitry V. Rutsky. Abstract: A. Ya. Zaslavskii's results on the existence of a linear (affine) selection for a linear (affine) or superlinear (convex) map $\Phi : K \to 2^Y$ defined on a convex cone (convex set) $K$ having the interpolation property are extended. We prove that they hold true under more general conditions on the values of the mapping and study some other properties of the selections. This leads to a characterization of Choquet simplexes in terms of the existence of continuous affine selections for arbitrary continuous convex maps. A few applications to analysis are given, including a construction that leads to the existence of a (not necessarily bounded) solution for the corona problem in polydisk $\mathbb D^n$ with radial boundary values that are bounded almost everywhere on $\mathbb T^n$. Archive classification: math.FA Submitted from: rutsky(a)pdmi.ras.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.3337 or http://arXiv.org/abs/1206.3337

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Abstract of a paper by Gabriele Bianchi, Almut Burchard, Paolo Gronchi, and Aljosa Volcic
by alspach＠math.okstate.edu 22 Jun '12

22 Jun '12

This is an announcement for the paper "Convergence in shape of Steiner symmetrizations" by Gabriele Bianchi, Almut Burchard, Paolo Gronchi, and Aljosa Volcic. Abstract: There are sequences of directions such that, given any compact set K in R^n, the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S^{n-1}.) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions. Archive classification: math.MG math.FA Mathematics Subject Classification: 52A40 (Primary) 28A75, 11K06, 26D15 (Secondary) Remarks: 11 pages Submitted from: gabriele.bianchi(a)unifi.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1206.2041 or http://arXiv.org/abs/1206.2041

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