PORTUGALIAE MATHEMATICA Vol. 60, No. 3, pp. 373377 (2003) 

Hypersurfaces of Infinite Dimensional Banach Spaces, Bertini Theorems and Embeddings of Projective SpacesE. BallicoDept. of Mathematics, University of Trento,38050 Povo (TN)  ITALY Email: ballico@science.unitn.it Abstract: Let $V$, $E$ be infinite dimensional Banach spaces, ${\bf{P}}(V)$ the projective space of all onedimensional linear subspaces of $V$, $W$ a finite codimensional closed linear subspace of ${\bf{P}}(V)$ and $X\subset{\bf{P}}(V)$ a closed analytic subset of finite codimension such that ${\bf{P}}(W)\subset X$ and $X$ is not a linear subspace of ${\bf{P}}(V)$. Here we show that $X$ is singular at some point of ${\bf{P}}(W)$. We also prove that any closed embedding $j:{\bf{P}}(V)\to{\bf{P}}(E)$ with $j({\bf{P}}(V))$ finite codimensional analytic subset of ${\bf{P}}(E)$ is a linear isomorphism onto a finite codimensional closed linear subspace of ${\bf{P}}(E)$. Keywords: infinitedimensional projective space; Banach analytic set; Banach analytic manifold; singular Banach analytic set; Berini theorem. Classification (MSC2000): 32K05, 58B12, 14N05. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
© 2003 Sociedade Portuguesa de Matemática
