We prove that, for a compact metric space X not reduced to a point, the existence of a bilinear mapping ⋄: C(X) × C(X) → C(X) satisfying ||f⋄g|| = ||f|| ||g|| for all f,g ∈ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-1-6,
author = {Antonio Moreno Galindo and \'Angel Rodr\'\i guez Palacios},
title = {A bilinear version of Holszty\'nski's theorem on isometries of C(X)-spaces},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {83-91},
zbl = {1129.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-1-6}
}
Antonio Moreno Galindo; Ángel Rodríguez Palacios. A bilinear version of Holsztyński's theorem on isometries of C(X)-spaces. Studia Mathematica, Tome 166 (2005) pp. 83-91. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-1-6/