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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 166 / 2005

Studia Mathematica 166 (2005), 83-91 MSC: Primary 46E15, 46B04; Secondary 46H70. DOI: 10.4064/sm166-1-6

#### Streszczenie

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.

#### Autorzy

• Antonio Moreno GalindoDepartamento de Análisis Matemático
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• Ángel Rodríguez PalaciosDepartamento de Análisis Matemático