On the Krein–Milman Property and the Bade Property

Armario, R.; García-Pacheco, Francisco Javier; Fernández, Francisco Javier Pérez

doi:10.1016/j.laa.2011.08.018

Cited by 5 publications

(13 citation statements)

References 11 publications

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“…The previous renorming in Equation (10) • M is that it is further from the original norm than • m , as we can see in Equation (11). Being further from the original norm makes the renorming lose plenty of geometric properties.…”

Section: Renormings Concerning Schauder Basesmentioning

confidence: 97%

“…It is trivial that • md defines a norm on X. Notice in first place that • m ≤ • M by Equation (11). By Equation 15,…”

Section: Renormings Concerning Schauder Basesmentioning

confidence: 99%

“…In fact, x + ∩N y = ∩N x+y for all x, y ∈ X. On the other hand, in [11], it was shown that W X := ∩N 0 is a closed vector subspace of X whose inherited topology is the trivial topology and which is topologically complemented with any of its linear complements. Any subspace Y of X verifying that Y ∩ W X = {0} is trivially Hausdorff.…”

Section: Impact Of Pre-schauder Bases On the Vector Topologymentioning

confidence: 99%

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Pre-Schauder Bases in Topological Vector Spaces

García-Pacheco

Fernández

2019

Symmetry

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A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

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Section: Renormings Concerning Schauder Basesmentioning

confidence: 97%

“…It is trivial that • md defines a norm on X. Notice in first place that • m ≤ • M by Equation (11). By Equation 15,…”

Section: Renormings Concerning Schauder Basesmentioning

confidence: 99%

Section: Impact Of Pre-schauder Bases On the Vector Topologymentioning

confidence: 99%

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Pre-Schauder Bases in Topological Vector Spaces

García-Pacheco

Fernández

2019

Symmetry

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“…Theorem 2.7 (Armario, García-Pacheco, and Pérez-Fernández, [1]). Let X be a real topological vector space.…”

Section: Convex Componentsmentioning

confidence: 99%

Convex components and multi-slices in real topological vector spaces

García-Pacheco¹

2015

Ann. Funct. Anal.

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It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent reformulation of the Krein-Milman property (involving drops among other objects). Finally, we study the nature of convex functions and provide some results on their support in order to introduce the concept of multi-slice, that is, slices determined by convex functions (instead of by linear functions). Among other things, we prove that the boundary of a closed convex set with non-empty interior can be obtained as the set of support points of a certain lower semi-continuous convex function on that convex set.

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“…One of the first assignments of Prof. García-Pacheco was to co-advice Rafael Armario together with Prof. Pérez-Fernández, continuing this way the work once started by Prof. Antonio Aizpuru. The fruit of this co-advising was the co-authorship of the papers [3, 4,12,13,40]. The present book consists of those papers.Many anecdotes occurred during his graduate school period.…”

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confidence: 99%

Vector-Valued Banach Limits and Vector-Valued Almost Convergence

Garcia-Pacheco,

Perez-Fernandez

2018

Preprint

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This book was intended to be Rafael Armario's Ph.D. thesis dissertation for obtaining his doctoral degree in Pure Mathematics at the University of Cádiz (Spain, EU). Unfortunately he passed away on January 28 th , 2013, right after he started writing this manuscript. His two Ph.D. advisors at the time of his death (the authors of this book) decided then to finish his work in book format and this is how this manuscript was given birth. Rafael Armario's lifeRafael Armario, better known as "Rafa" by his friends, was a joyful and warm person, a beloved son and friend, a very hard-working graduate student, an excellent highschool teacher, and an extremely brilliant mathematician. Childhood and early lifeRafa was born and raised in Cadiz, the oldest European city, within a humble middleclass family. He quickly developed a strong interest in the Cadiz culture and folklore, in particular, in the worldwide famous Cadiz carnival. Other local feasts such as the El Puerto fair also grew on him in a very profound way. University of Cadiz undergraduate periodRafa decided to register as a math major in the university of Cadiz after graduating from his neighborhood high school. He met a lot of friends there among students and i ii PREFACE professors, including the one who left the strongest imprint on him, Prof. Antonio Aizpuru, who eventually became his Ph.D. advisor. He also met a bunch of friends a buddies like Fernandito, Mari, Sole, Marina, Marimar, el Gafas, etc. Rafael Armario's workDuring his working period as a top-level mathematician, Rafa accomplished many objectives and obtained many different results. Indeed, all the theorems, definitions, examples, etc., which are anonymous in this book (including those of the framework) are due to his joint work with Prof. Aizpuru and the authors of this manuscript. University of Cadiz graduate periodRafa's graduate period was undoubtedly conditioned by the death of his Ph.D. advisor at the time, Prof. Antonio Aizpuru, who passed away in March 2008 after Rafa and him published their first paper (see [5]), which was also co-authored by his Ph.D. co-advisor at the time, Prof. Francisco Javier Pérez-Fernández.In September 2010, Prof. García-Pacheco, better known as Paquito, was hired at the level of Assistant Professor by the Mathematics Department of the University of Cadiz after spending nine semesters (three American academic years) at Kent State University (Ohio, USA) and ten semesters at Texas A&M University (Texas, USA). One of the first assignments of Prof. García-Pacheco was to co-advice Rafael Armario together with Prof. Pérez-Fernández, continuing this way the work once started by Prof. Antonio Aizpuru. The fruit of this co-advising was the co-authorship of the papers [3, 4,12,13,40]. The present book consists of those papers.Many anecdotes occurred during his graduate school period. For instance, in one occasion, Rafa and Paquito shared a hotel room, due to a lack of research funding, while attending a conference held in the university of Almeria (Spain, EU). The hotel wor...

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On the Krein–Milman Property and the Bade Property

Cited by 5 publications

References 11 publications

Pre-Schauder Bases in Topological Vector Spaces

Pre-Schauder Bases in Topological Vector Spaces

Convex components and multi-slices in real topological vector spaces

Vector-Valued Banach Limits and Vector-Valued Almost Convergence

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