A bilinear version of Holsztyński's theorem on isometries of $C(X)$-spaces

Antonio Moreno Galindo ¹   ; Ángel Rodríguez Palacios ¹

¹ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 166 (2005) no. 1, pp. 83-91

Résumé

We prove that, for a compact metric space $X$ not reduced to a point, the existence of a bilinear mapping $\diamond :C(X)\times C(X)\to C(X)$ satisfying $\Vert f\diamond g\Vert =\Vert f\Vert\, \Vert g\Vert $ for all $f,g\in C(X)$ is equivalent to the uncountability of $X$. This is derived from a bilinear version of Holsztyński's theorem [3] on isometries of $C(X)$-spaces, which is also proved in the paper.

Détail

DOI : 10.4064/sm166-1-6

Keywords: prove compact metric space reduced point existence bilinear mapping diamond times satisfying vert diamond vert vert vert vert vert equivalent uncountability derived bilinear version holszty skis theorem nbsp isometries spaces which proved paper