Abstract

The next result of the paper is motivated by a theorem of Alexander which says that a family of holomorphic maps from the unit ball in \(\mathbb C^n\) to \(\mathbb C\) is normal iff the restriction of it to every complex line \(l\) through 0 is normal. Namely, the author proves that a family \(\mathcal F\) of holomorphic maps of \(\mathbb B_n\subset\mathbb C^n\) into \(\mathbb P^N(\mathbb C)\) is uniformly Montel at \(0\in\mathbb B_n\) iff \(\mathcal F\left| _{l}\right.\) is uniformly Montel at \(0\in l\cap\mathbb B_n\) for all complex lines \(l\) through the origin. A sufficient condition for the normality of a family of holomorphic maps of a domain \(D\subset\mathbb C^n\) into \(\mathbb P^N(\mathbb C)\) is also given.