scholarly journals Some applications of model theory in Banach space theory

1976 ◽  
Vol 9 (1-2) ◽  
pp. 49-121 ◽  
Author(s):  
Jacques Stern
Keyword(s):  
Banach Space ◽  
Model Theory ◽  
Space Theory

scholarly journals Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Siran Li
Keyword(s):  
Banach Space ◽  
Discrete Geometry ◽  
Thin Shell ◽  
Euclidean Geometry ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Space Theory ◽  
Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

Three-space Problems in Banach Space Theory

1997 ◽  
Author(s):  
Jesús M. F. Castillo ◽  
Manuel González
Keyword(s):  
Banach Space ◽  
Space Theory ◽  
Three Space

Tensor Products of Operator Spaces II

1991 ◽  
Vol 44 (1) ◽  
pp. 75-90 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Elementary Theory ◽  
Tensor Products ◽  
Space Theory ◽  
Operator Spaces ◽  
Tensor Norm ◽  
Tensor Norms

AbstractTogether with Vern Paulsen we were able to show that the elementary theory of tensor norms of Banach spaces carries over to operator spaces. We suggested that the Grothendieck tensor norm program, which was of course enormously important in the development of Banach space theory, be carried out for operator spaces. Some of this has been done by the authors mentioned above, and by Effros and Ruan. We give alternative developments of some of this work, and otherwise continue the tensor norm program.

scholarly journals The Hahn Sequence Space Defined by the Cesáro Mean

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Murat Kirişci
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Matrix Transformations ◽  
Space Theory ◽  
Topological Properties ◽  
Cesàro Mean ◽  
First Order ◽  
Inclusion Relations ◽  
The Matrix ◽  
Cesaro Mean

The -space of all sequences is given as such that converges and is a null sequence which is called the Hahn sequence space and is denoted by . Hahn (1922) defined the space and gave some general properties. G. Goes and S. Goes (1970) studied the functional analytic properties of this space. The study of Hahn sequence space was initiated by Chandrasekhara Rao (1990) with certain specific purpose in the Banach space theory. In this paper, the matrix domain of the Hahn sequence space determined by the Cesáro mean first order, denoted by , is obtained, and some inclusion relations and some topological properties of this space are investigated. Also dual spaces of this space are computed and, matrix transformations are characterized.

scholarly journals Linear Hahn–Banach extension operators

1989 ◽  
Vol 32 (1) ◽  
pp. 53-57 ◽  
Author(s):  
Brailey Sims ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Model Theory ◽  
General Result ◽  
Extension Operator ◽  
Density Character ◽  
Finite Dimensional ◽  
Skolem Theorem

Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model 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>

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