scholarly journals On biorthogonal systems and Maxur's intersection property

2004 ◽  
Vol 69 (1) ◽  
pp. 107-111 ◽  
Author(s):  
Jan Rychtář
Keyword(s):  
Banach Spaces ◽  
Intersection Property ◽  
Sufficient Condition ◽  
Biorthogonal Systems ◽  
Markushevich Basis ◽  
Mazur's Intersection Property

We give a characterisation of Banach spaces X containing a subspace with a shrinking Markushevich basis {xγ, fγ}γ∈Γ. This gives a sufficient condition for X to have a renorming with Mazur's intersection property.

Un Théorème de Transfert Pour la Propriété des Boules

1987 ◽  
Vol 30 (3) ◽  
pp. 295-300 ◽  
Author(s):  
Robert Deville
Keyword(s):  
Banach Spaces ◽  
Equivalent Norm ◽  
Intersection Property ◽  
Mazur's Intersection Property

AbstractWe show that, if X and Y are Banach spaces such that X has the Mazur's intersection property and such that there exists T, an operator from Y into X so that T* and T** are injective, then there exists on Y an equivalent norm which has the Mazur's intersection property.We deduce from this result and from a result of M. Talagrand that there exists on the long James space J(η) an equivalent norm which has the Mazur's intersection property.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

On the Cones Associated with Biorthogonal Systems and Bases in Banach Spaces

1969 ◽  
Vol 21 ◽  
pp. 1206-1217 ◽  
Author(s):  
C. W. Mcarthur ◽  
Ivan Singer ◽  
Mark Levin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Biorthogonal System ◽  
Biorthogonal Systems ◽  
Image Position

1. Let E be a Banach space (by this we shall mean, for simplicity, a real Banach space) and (xn,fn) ({xn} ⊂ E, {fn} ⊂ E*) a biorthogonal system, such that {fn} is total on E (i.e. the relations x ∈ E,fn(x) = 0, n = 1, 2, …, imply x = 0). Then it is natural to consider the cone1which we shall call “the cone associated with the biorthogonal system (xn,fn)”. In particular, if {xn} is a basis of E and {fn} the sequence of coefficient functional associated with the basis {xn}, this cone is nothing else but2and we shall call it “the cone associated with the basis {xn}”.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

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