Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications

http://dx.doi.org/10.4153/CMB-2002-001-7
Canad. Math. Bull. 45(2002), 3-10

  Published:2002-03-01
 Printed: Mar 2002

We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.