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Petr Hájek - One of the best experts on this subject based on the ideXlab platform.
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an Asplund Space with norming marku v s evi v c basis that is not weakly compactly generated
arXiv: Functional Analysis, 2020Co-Authors: Petr Hájek, Tommaso Russo, Jacopo Somaglia, Stevo TodorcevicAbstract:We construct an Asplund Banach Space $\mathcal{X}$ with a norming Markusevic basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact Space, that also solves a question due to Wieslaw Kubiś and Arkady Leiderman.
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intersection property for Asplund Spaces
2013Co-Authors: Miroslav Bačák, Petr HájekAbstract:Abstract. The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin’s Maximum MM axiom), that every Asplund Space of density character ω1 has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund Spaces of density ω1 is undecidable in ZFC. The Mazur intersection property (MIP for short) was first investigated by S. Mazur in [9] as a purely geometrical isometric property of a Banach Space, and has since been studied extensively over the years. An early result of Mazur claims that a Banach Space with a Fréchet differentiable norm (necessarily an Asplund Space) has the MIP ([9]). Phelps [11] proved that a separable Banach Space has a MIP renorming if and only if its dual is separable, or equivalently, if it is an Asplund Space. Much of the further progress in the theory depended on an important characterization of MIP, due to Giles, Gregory and Sims [5], by the property that w∗-denting points of BX ∗ are norm dense in SX∗. This result again suggests a close connection of MIP to Asplund Spaces, as the latter can be characterized in a similar way as Spaces such that bounded subsets of their dual are w∗-dentable. It has opened a way to applying biorthogonal systems to the MIP. Namely, Jiménez and Moreno [8] have proved that if a Banach Space X ∗ admits a fundamental biorthogonal system {(xα, fα)}, where fα belong to X ⊂ X∗ ∗ , then X has a MIP renorming. As a corollary to this criterion ([8]), they get that every Banach Space can be embedded into a Banach Space which is MIP renormable, a rather surprising result which in particular strongly shows that MIP and Asplund properties, although closel
Giles J. R. - One of the best experts on this subject based on the ideXlab platform.
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A continuity characterization of Asplund Spaces
Cambridge University Press, 2011Co-Authors: Giles J. R.Abstract:A Banach Space is an Asplund Space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler
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Fréchet intermediate differentiability of Lipschitz functions on Asplund Spaces
Cambridge University Press, 2009Co-Authors: Giles J. R.Abstract:The deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund Space is densely Fréchet differentiable. However, the simpler Fabian-Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated Spaces
Fuchun Yang - One of the best experts on this subject based on the ideXlab platform.
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generalized euler identity for subdifferentials of homogeneous functions and applications
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Fuchun YangAbstract:In this paper, we mainly consider subdifferentials and basic subdifferentials of homogeneous functions defined on real Banach Space and Asplund Space respectively, and obtain the generalized Euler identity. As applications, we consider constrained optimization problems and several geometric properties of Banach Space.
Hájek Petr - One of the best experts on this subject based on the ideXlab platform.
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An Asplund Space with norming Marku\v{s}evi\v{c} basis that is not weakly compactly generated
2020Co-Authors: Hájek Petr, Russo Tommaso, Somaglia Jacopo, Todorčević StevoAbstract:We construct an Asplund Banach Space $\mathcal{X}$ with a norming Marku\v{s}evi\v{c} basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact Space, that also solves a question due to Wies\l aw Kubi\'s and Arkady Leiderman.Comment: 20 p
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Note on Kadets Klee property and Asplund Spaces
2013Co-Authors: Hájek Petr, Talponen JarnoAbstract:A typical result in this note is that if $X$ is a Banach Space which is a weak Asplund Space and has the $\omega^*$-$\omega$-Kadets Klee property, then $X$ is already an Asplund Space
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Mazur intersection property for Asplund Spaces
Elsevier Inc., 2008Co-Authors: Bačák Miroslav, Hájek PetrAbstract:AbstractThe main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund Space of density character ω1 has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund Spaces of density ω1 is undecidable in ZFC
Galicer Daniel - One of the best experts on this subject based on the ideXlab platform.
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The symmetric Radon–Nikodým property for tensor norms
Elsevier Inc., 2011Co-Authors: Carando Daniel, Galicer DanielAbstract:AbstractWe introduce the symmetric Radon–Nikodým property (sRN property) for finitely generated s-tensor norms β of order n and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if β is a projective s-tensor norm with the sRN property, then for every Asplund Space E, the canonical mapping ⊗˜βn,sE′→(⊗˜β′n,sE)′ is a metric surjection. This can be rephrased as the isometric isomorphism Qmin(E)=Q(E) for some polynomial ideal Q. We also relate the sRN property of an s-tensor norm with the Asplund or Radon–Nikodým properties of different tensor products. As an application, results concerning the ideal of n-homogeneous extendible polynomials are obtained, as well as a new proof of the well-known isometric isomorphism between nuclear and integral polynomials on Asplund Spaces. An analogous study is carried out for full tensor products
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The symmetric Radon-Nikod\'ym property for tensor norms
'Elsevier BV', 2010Co-Authors: Carando Daniel, Galicer DanielAbstract:We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $\beta$ is a projective s-tensor norm with the sRN property, then for every Asplund Space $E$, the canonical map $\widetilde{\otimes}_{\beta}^{n,s} E' \to \Big(\widetilde{\otimes}_{\beta'}^{n,s} E \Big)'$ is a metric surjection. This can be rephrased as the isometric isomorphism $\mathcal{Q}^{min}(E) = \mathcal{Q}(E)$ for certain polynomial ideal $\Q$. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties of different tensor products. Similar results for full tensor products are also given. As an application, results concerning the ideal of $n$-homogeneous extendible polynomials are obtained, as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on Asplund Spaces.Comment: 17 page
