Proximinal subspaces of \(C(\mathcal Q)\) of finite codimension
The structure of a proximinal subspace \(G\) of \(C(\mathcal Q)\) of codimension \(n\), \(\mathcal Q\) being a compact Hausdorff space, is studied in terms of the geometry of the range of the vector measure \(\nu=(\nu_1, \dots, \nu_n)\), where \(\{\nu_1, \dots, \nu_n\}\) is a basis for the annihilator \(G^\perp\). In particular, if \(\nu\) is non-atomic, \(G\) is proximinal if and only if for every \(P\in \text{Ext}\,R(v)\) there exists a clopen subset \(C\) of \(\bigcup_{i=1}^n S(\nu_i)\) such that \(\nu(C)=P\). It is shown that the annihilator of a proximinal finitely complemented subspace of \(C(\mathcal Q)\) is a finite dimensional subspace of \(C(\mathcal Q)\). A set \(G\subset C(\mathcal Q)\) of codimension 2 is proximinal if and only if every extreme point of the range of the measure \((\nu_1,\nu_2)\) is the image under \((\nu_1,\nu_2)\) of a clopen subset of \(S(\nu_1)\cup S(\nu_2)\).
scientific article; zbMATH DE number 1389111
scientific article