Analytic Subsets of Hilbert Spaces †

We show that every complete metric space is homeomorphic to the locus of zeros of an entire analytic map from a complex Hilbert space to a complex Banach space. As a corollary, every separable complete metric space is homeomorphic to the locus of zeros of an entire analytic map between two complex Hilbert spaces. §1. Douady had observed [8] that every compact metric space is homeomorphic to the locus of zeros of an analytic (in fact, continuous polynomial) map between two suitable Banach spaces. In other words, every compact metric space is homeomorphic to an algebraic subset of a complex Banach space. Proving Douady’s conjecture, the present author has shown [13] that every complete metric space is isometric to an algebraic subset of a complex Banach space. Hilbert spaces provide a particularly favourable setting for Banach analytic geometry (cf. [16]), and the following question suggested by Norm Dancer is very natural: what can be said about the topology of analytic subsets of Hilbert (or just reflexive Banach) spaces? A previously known result belongs to Ramis [15] who embedded the Cantor set topologically in a Hilbert space as an algebraic subset. Main Theorem. Every complete metric space is homeomorphic to the precise locus of zeros of an entire analytic map from a complex Hilbert space to a complex Banach space. Call a Banach analytic space [7] an Hilbert analytic space if it is modelled on analytic subsets of an Hilbert space (zeros of Banach-valued analytic maps). The following corollary betters both Douady’s and the present author’s earlier results. Corollary 1. A paracompact topological space admits the structure of an Hilbert analytic space if and only if it is metrizable with a complete metric. Corollary 2. Every separable complete metric space is homeomorphic to the locus of zeros of an entire map between two separable complex Hilbert spaces. Every closed subset of a separable real Hilbert space is the precise locus of zeros of a C functional (see e.g. [9], 2.C), but obviously this result does not extend to real analytic functionals. However, one has: 1991 Mathematics Subject Classification. 32K05, 58B12. † Research Report RP-95-157, Department of Mathematics, Victoria University of Wellington, March 1995.