scholarly journals Banach spaces and Ramsey Theory: some open problems

2010 ◽  
Vol 104 (2) ◽  
pp. 435-450 ◽  
Author(s):  
P. Dodos ◽  
J. Lopez-Abad ◽  
S. Todorcevic
Keyword(s):  
Banach Spaces ◽  
Ramsey Theory ◽  
Open Problems

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Errata and addendum: tensor products and Dunford–Pettis sets

2007 ◽  
Vol 143 (2) ◽  
pp. 387-390
Author(s):  
Ioana Ghenciu ◽  
Paul Lewis
Keyword(s):  
Banach Spaces ◽  
Ramsey Theory ◽  
Tensor Products ◽  
Schauder Basis ◽  
Fundamental Result ◽  
Individual Basis ◽  
Quick Proof

AbstractGhenciu and Lewis introduced the notion of a strong Dunford–Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Elton's results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis.

Open Problems in Euclidean Ramsey Theory

Ramsey Theory ◽  
2011 ◽  
pp. 115-120
Author(s):  
Ron Graham ◽  
Eric Tressler
Keyword(s):  
Ramsey Theory ◽  
Open Problems ◽  
Euclidean Ramsey Theory

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals Open problems in Banach spaces and measure theory

2017 ◽  
Vol 40 (6) ◽  
pp. 811-832
Author(s):  
José Rodríguez
Keyword(s):  
Banach Spaces ◽  
Measure Theory ◽  
Open Problems

Open Problems Column Edited by William Gasarch This Issue's Column!

2021 ◽  
Vol 51 (4) ◽  
pp. 30-46
Author(s):  
William Gasarch
Keyword(s):  
Elementary Proof ◽  
Ramsey Theory ◽  
Open Problems ◽  
General Theme ◽  
Open Questions ◽  
Ramsey Type

In this column we state a class of open problems in Ramsey Theory. The general theme is to take Ramsey-type statements that are false and weaken them by allowing the homogenous set to use more than one color. This concept is not new, and the theorems we state and/or prove are not new; however, the open questions that request easier proofs of the known theorems (or weaker versions) may be new. We use the phrase an elementary proof. This is not meant to be a technical or rigorous term. What we really mean is a proof that can be taught in an undergraduate combinatorics course. A good example of what we mean is the proof of Theorem 9.3.

A Ramsey Theorem with an Application to Sequences in Banach Spaces

2012 ◽  
Vol 55 (2) ◽  
pp. 410-417
Author(s):  
Robert Service
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Ramsey Theory ◽  
Basic Sequence ◽  
Alternative Proof ◽  
Ramsey Theorem ◽  
Combinatorial Lemma

AbstractThe notion of a maximally conditional sequence is introduced for sequences in a Banach space. It is then proved using Ramsey theory that every basic sequence in a Banach space has a subsequence which is either an unconditional basic sequence or a maximally conditional sequence. An apparently novel, purely combinatorial lemma in the spirit of Galvin's theorem is used in the proof. An alternative proof of the dichotomy result for sequences in Banach spaces is also sketched, using the Galvin–Prikry theorem.

Sign in / Sign up

Export Citation Format

Share Document