Douady's conjecture on Banach analytic spaces

TL;DR

This paper proves Douady's conjecture that every complete metric space can be realized as the zero set of an analytic map between Banach spaces, establishing a fundamental link between topology and Banach analytic spaces.

Contribution

It confirms Douady's long-standing conjecture, showing complete metric spaces are homeomorphic and isometric to zero loci of Banach space analytic maps, and characterizes Banach analytic spaces topologically.

Findings

Complete metric spaces are homeomorphic to zero sets of Banach analytic maps.

A paracompact topological space is a Banach analytic space iff it is metrizable with a complete metric.

The result provides a topological characterization of Banach analytic spaces.

Abstract

We show that, as conjectured by Adrien Douady back in 1972, every complete metric space is homeomorphic (moreover, isometric) to the locus of zeros of an analytic map between two Banach spaces. As a corollary, a paracompact topological space admits the structure of a Banach analytic space if and only if it is metrizable with a complete metric.