IJPAM: Volume 21, No. 4 (2005)

ON THE ZERO-LOCUS OF REAL ANALYTIC
FUNCTIONS ON DOMAINS OF TOPOLOGICAL
VECTOR SPACES

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Let $V$ be a real Fréchet space without a continuos norm, $W$ a real Banach space, $U$ an open subset of $V$ and $f: U \to W$ a real analytic function. Then for every $P\in f^{-1}(0)$ there is no open subset $\Omega$ of $P$ in $U$ and a closed finite-dimension real submanifold $Z$ of $\Omega$ such that $f^{-1}(0)\cap \Omega \subseteq Z$. Furthermore, there is an open neighborhood $A$ of $P$ such for every integer $z\ge 1$, there is an a $z$-dimensional closed real submanifold $T_z$ of $A$ such that $P\in T_z \subset f^{-1}(0)$.

Received: May 6, 2005

AMS Subject Classification: 32C05, 32D20, 46E99

Key Words and Phrases: real analytic function, real analytic function in infinite-dimensional topological vector spaces, topological vector space without a continuous norm

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 21
Issue: 4