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

We prove that every infinite-dimensional Banach space X having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to X \ {0}. More generally, if X is an infinitedimensional 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 diffeomorphic. 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 to get diffeomorphisms instead of merely homeomorphisms. In 1966, C. Bessaga [5] proved that if X is an infinite-dimensional Hilbert space then X is C∞ diffeomorphic to both X \ {0} and its unit sphere. Some twelve years later, the second-named author [16] developed the so-called non-complete norm technique of Bessaga’s in the smooth case and showed that if X has a non-complete C p smooth norm then X and X \ K are C p diffeomorphic for any compact set K ⊂ X. Unfortunately, it is not known whether every infinite-dimensional space with an equivalent C p smooth norm must have a non-complete C p smooth norm too, so that this result does not allow us to conclude that the same holds true for all infinitedimensional Banach spaces with smooth norms. Without proving the existence of smooth non-complete norms, the first-named author recently showed [2] that every infinite-dimensional Banach space with a (not necessarily equivalent) C p smooth norm is C p diffeomorphic to X \ {0} and, furthermore, that every hyperplane in X is C p diffeomorphic to the sphere {x ∈ X | (x) = 1}. Then the present authors strengthened the asymmetric norm technique of deleting points introduced in [2] so as to obtain very general results concerning smooth negligibility of compact sets and subspaces [3]. These results allow to enlarge the class of spaces in which some striking applications of negligibility theory are valid (see [3], [4], [7], [19], [20], [21], [25]). Received by the editors January 20, 2000; revised March 14, 2000. AMS subject classification: 46B20, 58B99. c ©Canadian Mathematical Society 2002.