Banach

banach@mathdept.okstate.edu

2549 discussions

Abstract of a paper by Taras Banakh, Joanna Garbulińska-Węgrzyn
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “The universal separable Banach space with a monotone unordered basis” by Taras Banakh<https://arxiv.org/find/math/1/au:+Banakh_T/0/1/0/all/0/1>, Joanna Garbulińska-Węgrzyn<https://arxiv.org/find/math/1/au:+Garbulinska_Wegrzyn_J/0/1/0/all/0/1>. Abstract: Using the technique of Fra\"iss\'e theory we construct a universal object in the class of separable Banach spaces with normalized monotone unordered Schauder basis. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.07433

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Abstract of a paper by Antonio Avilés, Grzegorz Plebanek, José David Rodríguez Abellán
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Chain conditions in free Banach lattices” by Antonio Avilés<https://arxiv.org/find/math/1/au:+Aviles_A/0/1/0/all/0/1>, Grzegorz Plebanek<https://arxiv.org/find/math/1/au:+Plebanek_G/0/1/0/all/0/1>, José David Rodríguez Abellán<https://arxiv.org/find/math/1/au:+Abellan_J/0/1/0/all/0/1>. Abstract: We show that the free Banach lattice FBL$[E]$ generated by a Banach space $E$ satisfies the countable chain condition, and if the Banach space $E$ is moreover weakly compactly generated, then FBL$[E]$ satisifes Knaster's condition $K_2$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.07027

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Abstract of a paper by Francisco Javier Garcia-Pacheco, Francisco Javier Perez-Fernandez
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Vector-Valued Banach Limits and Vector-Valued Almost Convergence” by Francisco Javier Garcia-Pacheco<https://arxiv.org/find/math/1/au:+Garcia_Pacheco_F/0/1/0/all/0/1>, Francisco Javier Perez-Fernandez<https://arxiv.org/find/math/1/au:+Perez_Fernandez_F/0/1/0/all/0/1>. Abstract: This book pretends to compile the latest advances on vector-valued Banach limits as well as their applications to vector-valued almost convergence. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.06235

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Abstract of a paper by Eftychios Glakousakis, Sophocles Mercourakis
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Antipodal sets in infinite dimensional Banach spaces” by Eftychios Glakousakis<https://arxiv.org/find/math/1/au:+Glakousakis_E/0/1/0/all/0/1>, Sophocles Mercourakis<https://arxiv.org/find/math/1/au:+Mercourakis_S/0/1/0/all/0/1>. Abstract: The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)$−separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subset S_X$ and a constant$d>1$, satisfying the property that for every $x, y\in S$ with $x\neq y$ there exists $f\in B_{X^*}$ such that $d\leq f(x)-f(y)$ and $f(y)\leq f(z)\leq f(x)$, for all $z\in S$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.02002

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Abstract of a paper by Saak Gabriyelyan
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Locally convex spaces and Schur type properties” by Saak Gabriyelyan<https://arxiv.org/find/math/1/au:+Gabriyelyan_S/0/1/0/all/0/1>. Abstract: We extend Rosenthal's characterization of Banach spaces with the Schur property to a wide class of locally convex spaces (lcs) strictly containing the class of Fr\'{e}chet spaces by showing that for an lcs $E$ from this class the following conditions are equivalent: (1) $E$ has the Schur property, (2) $E$ and $E_w$ have the same sequentially compact sets, where $E_w$ is the space $E$ with the weak topology, (3) $E$ and $E_w$ have the same compact sets, (4) $E$ and $E_w$ have the same countably compact sets, (5) $E$ and $E_w$ have the same pseudocompact sets, (6) $E$ and $E_w$ have the same functionally bounded sets, (7) every bounded non-precompact sequence in $E$ has a subsequence which is equivalent to the unit basis of $\ell_1$. We show that for a quasi-complete lcs conditions (3)-(6) are equivalent to (8) every non-precompact bounded subset of $E$ has an infinite subset which is discrete and $C$-embedded in $E_w$. We prove that every real locally convex space is a quotient space of an lcs $E$ satisfying conditions (1)-(5). The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.01992

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Abstract of a paper by Ryan M. Causey
by Bentuo Zheng (bzheng) 03 Feb '18

03 Feb '18

This is an announcement for the paper “Concerning $q$-summable Szlenk index” by Ryan M. Causey<https://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>. Abstract: For each ordinal $\xi$ and each $1\leq q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^*$-compact set a transfinite, asymptotic analogue $a_{\xi, p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $a_{\xi, p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $a_{\xi, p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $a_{\xi, p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $a_{\xi, p}$ seminorms under $\ell_r$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_p$ and $c_0$ direct sums of operators. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.00033

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Abstract of a paper by Sofiya Ostrovska, Mikhail I. Ostrovskii
by Bentuo Zheng (bzheng) 02 Jan '18

02 Jan '18

This is an announcement for the paper “On embeddings of locally finite metric spaces into $\ell_p$” by Sofiya Ostrovska<https://arxiv.org/find/math/1/au:+Ostrovska_S/0/1/0/all/0/1>, Mikhail I. Ostrovskii<https://arxiv.org/find/math/1/au:+Ostrovskii_M/0/1/0/all/0/1>. Abstract: It is known that if finite subsets of a locally finite metric space $M$ admit $C$-bilipschitz embeddings into $\ell_p$ $(1\leq p\leq\infty)$, then for every $\epsilon>0$, the space $M$ admits a $C+\epsilon$-bilipschitz embedding into $\ell_p$. The goal of this paper is to show that for $p\neq 2, \infty$ this result is sharp in the sense that ϵ cannot be dropped out of its statement. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1712.08255

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Abstract of a paper by Fernando Albiac, José L. Ansorena
by Bentuo Zheng (bzheng) 02 Jan '18

02 Jan '18

This is an announcement for the paper “Building highly conditional quasi-greedy bases in classical Banach spaces” by Fernando Albiac<https://arxiv.org/find/math/1/au:+Albiac_F/0/1/0/all/0/1>, José L. Ansorena<https://arxiv.org/find/math/1/au:+Ansorena_J/0/1/0/all/0/1>. Abstract: It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_m[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_m[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m=\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. However, in the existing literature one finds very few instances of non-superreflexive spaces possessing quasi-greedy basis with conditionality constants as large as possible. Our goal in this article is to fill this gap. To that end we enhance and exploit a combination of techniques developed independently, on the one hand by Garrig\'os and Wojtaszczyk in [Conditional quasi-greedy bases in Hilbert and Banach spaces, Indiana Univ. Math. J. 63 (2014), no. 4, 1017-1036] and, on the other hand, by Dilworth et al. in [On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101], and craft a wealth of new examples of non-superreflexive classical Banach spaces having quasi-greedy bases $\mathcal{B}$ with $k_m[\mathcal{B}]=\mathcal{O}(\log m)$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1712.04004

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Abstract of a paper by Jesús A. Jaramillo, Sebastián Lajara, Óscar Madiedo
by Bentuo Zheng (bzheng) 02 Jan '18

02 Jan '18

This is an announcement for the paper “Inversion of nonsmooth maps between Banach spaces” by Jesús A. Jaramillo<https://arxiv.org/find/math/1/au:+Jaramillo_J/0/1/0/all/0/1>, Sebastián Lajara<https://arxiv.org/find/math/1/au:+Lajara_S/0/1/0/all/0/1>, Óscar Madiedo<https://arxiv.org/find/math/1/au:+Madiedo_O/0/1/0/all/0/1>. Abstract: We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1712.00565

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Abstract of a paper by Marek Cúth, Benjamin Vejnar, Ondřej Kurka
by Bentuo Zheng (bzheng) 02 Jan '18

02 Jan '18

This is an announcement for the paper “Large separated sets of unit vectors in Banach spaces of continuous functions” by Marek Cúth<https://arxiv.org/find/math/1/au:+Cuth_M/0/1/0/all/0/1>, Benjamin Vejnar<https://arxiv.org/find/math/1/au:+Vejnar_B/0/1/0/all/0/1>, Ondřej Kurka<https://arxiv.org/find/math/1/au:+Kurka_O/0/1/0/all/0/1>. Abstract: The paper is concerned with the problem whether a nonseparable $\C(K)$ space must contain a set of unit vectors whose cardinality equals to the density of $\C(K)$ such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of $\C(K)$ spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly $2$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1712.00478

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