Hypersurfaces of Infinite Dimensional Banach Spaces, Bertini Theorems and Embeddings of Projective Spaces

E. Ballico

Abstract: Let $V$, $E$ be infinite dimensional Banach spaces, ${\bf{P}}(V)$ the projective space of all one-dimensional 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: infinite-dimensional projective space; Banach analytic set; Banach analytic manifold; singular Banach analytic set; Berini theorem.

Classification (MSC2000): 32K05, 58B12, 14N05.

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