 
Summary: Canad. Math. Bull. Vol. 45 (1), 2002 pp. 310
RealAnalytic Negligibility of Points and
Subspaces in Banach Spaces, with
Applications
D. Azagra and T. Dobrowolski
Abstract. We prove that every infinitedimensional Banach space X having a (not necessarily equiva
lent) realanalytic norm is realanalytic diffeomorphic to X \ {0}. More generally, if X is an infinite
dimensional Banach space and F is a closed subspace of X such that there is a realanalytic seminorm
on X whose set of zeros is F, and X/F is infinitedimensional, then X and X \ F are realanalytic dif
feomorphic. As an application we show the existence of realanalytic free actions of the circle and the
ntorus on certain Banach spaces.
In 1951 Victor Klee proved that, if X is either a nonreflexive Banach space or
an infinitedimensional Lp
space and K is a compact subset of X then X \ K and X
are homeomorphic. He also showed that every infinitedimensional Hilbert space is
homeomorphic to its unit sphere, and he gave a complete topological classification
of the convex bodies of a Hilbert space. These results were later extended to the class
of all infinitedimensional Banach spaces by Bessaga and Klee (cf. [6], [8], [9], [10]).
If a subset A of X has the property that X and X \ A are homeomorphic, we say that
A is negligible. It is natural to ask whether this type of results can be sharpened so as
