Let Q be a compact set, and let C(Q) be the space of real valued continuous function on Q. In this paper we investigate the structure of a proximinal subspace G of C(Q) of codimension 2 in terms of the geometry of the range of the bidimensional measure m whose components constitute a a basis for the annihilator of G. In particular, we prove that if m is non-atomic, G is proximinal if and only if for every P which is extremal for the range of m there exists a clopen subset C of the support of m such that m(C)=P. The result is obtained combining two results: Garkavi’s characterization of proximinal subspaces of finite codimension in terms of the existence of special Hahn decompositions, and of a Radon-Nikodym type Theorem due to Greco.
Proximinal subspaces of C(Q) of finite codimension
MARTELLOTTI, Anna
1999
Abstract
Let Q be a compact set, and let C(Q) be the space of real valued continuous function on Q. In this paper we investigate the structure of a proximinal subspace G of C(Q) of codimension 2 in terms of the geometry of the range of the bidimensional measure m whose components constitute a a basis for the annihilator of G. In particular, we prove that if m is non-atomic, G is proximinal if and only if for every P which is extremal for the range of m there exists a clopen subset C of the support of m such that m(C)=P. The result is obtained combining two results: Garkavi’s characterization of proximinal subspaces of finite codimension in terms of the existence of special Hahn decompositions, and of a Radon-Nikodym type Theorem due to Greco.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
