IJPAM: Volume 27, No. 2 (2006)

TOPOLOGICAL VECTOR SPACES WITHOUT
REAL ANALYTIC ``$F$-NORMS''

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: [email protected]


Abstract.Here we point out the following consequences of a previous result of mine. Let $V$ be a Hausdorff and locally convex topological vector space without any continuous norm and $\vert \ \vert _F$ an $F$-norm defining the topology of $V$. Then there is no pair $(U,f)$ with $U$ an open neighborhood of $0\in V$ and $f: U \to \mathbb{R}$ a real analytic function such that $f(0) = 0$ and the sets $\{f < c\}_{c>0}$ form a fundamental system of open neighborhoods of $0$ in $V$. Furthermore, there is no strictly increasing function $g: [0,+\infty ) \to [0,+\infty )$ such that $g\circ \vert \ \vert _F: V \to [0,+\infty )$ is real analytic.

Received: November 29, 2005

AMS Subject Classification: 32K05, 32D20, 46E99

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

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 27
Issue: 2