scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

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.

scholarly journals Proximinal subspaces ofA(K)of finite codimension

2003 ◽  
Vol 2003 (39) ◽  
pp. 2501-2505
Author(s):  
T. S. S. R. K. Rao
Keyword(s):  
Compact Subset ◽  
Convex Set ◽  
Compact Convex ◽  
Continuous Functions ◽  
Finite Codimension ◽  
Sufficient Condition ◽  
Choquet Simplex ◽  
Necessary And Sufficient Condition ◽  
Compact Convex Set ◽  
Necessary And Sufficient

We study an analogue of Garkavi's result on proximinal subspaces ofC(X)of finite codimension in the context of the spaceA(K)of affine continuous functions on a compact convex setK. We give an example to show that a simple-minded analogue of Garkavi's result fails for these spaces. WhenKis a metrizable Choquet simplex, we give a necessary and sufficient condition for a boundary measure to attain its norm onA(K). We also exhibit proximinal subspaces of finite codimension ofA(K)when the measures are supported on a compact subset of the extreme boundary.

scholarly journals The extremal points of the range of a vector-valued measure

1972 ◽  
Vol 13 (1) ◽  
pp. 61-63 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Convex Set ◽  
Compact Convex ◽  
Extremal Points ◽  
Compact Convex Set ◽  
Vector Valued

Recently, several papers have investigated conditions under which the range of a vectorvalued measure is a compact convex set (see e.g. [1], [2], [3]). It therefore seems of interest to characterise the extremal points of the range in such cases.

Transitivity in spaces of vector-valued functions

2010 ◽  
Vol 53 (3) ◽  
pp. 601-608 ◽  
Author(s):  
Félix Cabello Sánchez
Keyword(s):  
Banach Space ◽  
Maximal Ideal ◽  
Real Banach Space ◽  
Cantor Set ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractWe exhibit a real Banach space M such that C(K,M) is almost transitive if K is the Cantor set, the growth of the integers in its Stone–Čech compactification or the maximal ideal space of L∞. For finite K, the space C(K,M) = M|K| is even transitive.

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