scholarly journals On a conjecture of Lindenstrauss and Perles in at most 6 dimensions

1978 ◽  
Vol 19 (1) ◽  
pp. 87-97 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Banach Space ◽  
Extreme Points ◽  
Linear Operators ◽  
Dimensional Banach Space ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space

In [1] J. Lindenstrauss and M. A. Perles studied the extreme points of the set of all linear operators T of norm ≤ 1 from a finite dimensional Banach space X into itself. In particular they studied the question “When do these extreme points form a semigroup?”.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

On Benson’s Definition of Area in Minkowski Space

1999 ◽  
Vol 42 (2) ◽  
pp. 237-247 ◽  
Author(s):  
A. C. Thompson
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Minkowski Space ◽  
Surface Area ◽  
Isoperimetric Problem ◽  
Dimensional Banach Space ◽  
Finite Dimensional ◽  
The One ◽  
Definition Of ◽  
Finite Dimensional Banach Space

AbstractLet (X, ‖ . ‖) be a Minkowski space (finite dimensional Banach space) with unit ball B. Various definitions of surface area are possible in X. Here we explore the one given by Benson [1], [2]. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

scholarly journals On the semigroup of Ck selfmaps of Rn

1974 ◽  
Vol 17 (4) ◽  
pp. 389-393
Author(s):  
G. R. Wood
Keyword(s):  
Banach Space ◽  
Topological Spaces ◽  
Dimensional Banach Space ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Montel Space ◽  
Fréchet Differentiable ◽  
Finite Dimensional Banach Space

Magill, Jr. and Yamamuro have been responsible in recent years for a number of papers showing that the property that every automorphism is inner is held by many semigroups of functions and relations on topological spaces. Following [9], we say a semigroup has the Magill property if every automorphism is inner. we say a semigroup has the Magill property if every automorphism is inner. That the semigroup of Fréchet differentiable selfmaps, D of a finite dimensional Banach space E, had the Magill property was shown in [10], while a lengthy result in [6] extended this to the semigroup of k times Fréchet differentiable selfmaps, Dk, of a Fréchet Montel space (FM-space). In the latter paper it was noted that with a little additional effort the semigroup Ck, of k times continuously Fréchet differentiable selfmaps of FM-space, could be shown to possess the Magill property. It is the purpose of this paper to present a simpler proof of this result in the case where the underlying space is finite dimensional.

A Simple Deviation from Relativity

1993 ◽  
Vol 48 (8-9) ◽  
pp. 932-934
Author(s):  
Georg Süssmann
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Special Relativity ◽  
Natural Generalization ◽  
Optical Methods ◽  
Dimensional Banach Space ◽  
Finite Dimensional ◽  
Quantum Optical ◽  
Finite Dimensional Banach Space ◽  
Space Concept

Abstract A test of special relativity is proposed by conceiving a rather natural generalization of the minkowskian spacetime. This is mathematically similar to generalizing the notion of finite dimensional Banach space of the related Hilbert space concept. A corresponding experiment might be feasible with appropriate quantum optical methods.

scholarly journals ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

2011 ◽  
Vol 53 (3) ◽  
pp. 443-449 ◽  
Author(s):  
ANTONÍN SLAVÍK
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Characteristic Property ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Counter Example ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Dvoretzky’S Theorem ◽  
The Given

AbstractThis paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

Near Triangularizability Implies Triangularizability

2004 ◽  
Vol 47 (2) ◽  
pp. 298-313 ◽  
Author(s):  
Bamdad R. Yahaghi
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Compact Operators ◽  
Linear Operators ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Ground Field ◽  
Subspace Theorem ◽  
Finite Dimensional

AbstractIn this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is or .

Geometry of Modulus Spaces

2002 ◽  
Vol 9 (2) ◽  
pp. 295-301
Author(s):  
R. Khalil ◽  
D. Hussein ◽  
W. Amin
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Linear Space ◽  
Extreme Points ◽  
Linear Operators ◽  
Modulus Function ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Increasing Function

Abstract Let ϕ be a modulus function, i.e., continuous strictly increasing function on [0, ∞), such that ϕ(0) = 0, ϕ(1) = 1, and ϕ(𝑥+𝑦) ≤ ϕ(𝑥)+ϕ(𝑦) for all 𝑥, 𝑦 in [0, ∞). It is the object of this paper to characterize, for any Banach space 𝑋, extreme points, exposed points, and smooth points of the unit ball of the metric linear space ℓ ϕ (𝑋), the space of all sequences (𝑥𝑛), 𝑥𝑛 ∈ 𝑋, 𝑛 = 1, 2, . . . , for which ∑ϕ(‖𝑥𝑛‖) < ∞. Further, extreme, exposed, and smooth points of the unit ball of the space of bounded linear operators on ℓ𝑝, 0 < 𝑝 < 1, are characterized.

scholarly journals On the set of limits of Riemann sums

1975 ◽  
Vol 20 (1) ◽  
pp. 59-65
Author(s):  
David Daniel ◽  
H. W. Ellis
Keyword(s):  
Banach Space ◽  
Extreme Points ◽  
Complete Characterization ◽  
Riemann Sums ◽  
Finite Dimensional

Let F map [0, 1] into a Banach space B and let R(F) denote the set of all limits of Riemann sums of F. The set R(F) need not be convex in general (Nakamura and Amemiya (1966)) but is always convex when B is finite dimensional as first shown by Hartman (1947). A proof of Hartman's result, based on a description of R(F) when the range of F is finite, was given in Ellis (1959). In this note this description is refined, the extreme points of R(F) are determined and the following complete characterization of R(F) is obtained (where Nn = {1,2, …, n}).

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