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Canad. Math. Bull. Vol. 45 (1), 2002 pp. 310 Real-Analytic Negligibility of Points and

Summary: Canad. Math. Bull. Vol. 45 (1), 2002 pp. 3­10
Real-Analytic Negligibility of Points and
Subspaces in Banach Spaces, with
D. Azagra and T. Dobrowolski
Abstract. We prove that every infinite-dimensional Banach space X having a (not necessarily equiva-
lent) real-analytic norm is real-analytic 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 real-analytic seminorm
on X whose set of zeros is F, and X/F is infinite-dimensional, then X and X \ F are real-analytic dif-
feomorphic. As an application we show the existence of real-analytic free actions of the circle and the
n-torus on certain Banach spaces.
In 1951 Victor Klee proved that, if X is either a non-reflexive Banach space or
an infinite-dimensional Lp
space and K is a compact subset of X then X \ K and X
are homeomorphic. He also showed that every infinite-dimensional 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 infinite-dimensional 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


Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


Collections: Mathematics