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Hilbert space factor of metric spaces

Thomas Foertsch, Alexander Lytchak, Elefterios Soultanis

TL;DR

The paper proves that every complete metric space X decomposes uniquely as X = Y × H, where H is a (possibly finite- or zero-dimensional) Hilbert space and Y contains no splitting lines. It introduces the notion of factor subsets and a Zorn-lemma argument to construct a maximal Hilbert-factor H_x through any point x, then shows the complementary factor Y has no splitting lines and that H_x equals the union of all splitting lines through x. This yields a canonical isometry decomposition with Iso(X) ≅ Iso(Y) × Iso(H). The results connect to de Rham-type decompositions and extend to various geometric contexts, including CAT(0) and Alexandrov spaces, by characterizing the Hilbert-factor via splitting lines and projections. The method hinges on the interplay between factor subsets, their intersections, and Zorn’s Lemma to secure maximal Hilbert-factors.

Abstract

We prove that any complete metric space has a unique decomposition as a direct product of a possibly finite or zero-dimensional Hilbert space and a space that does not split off lines.

Hilbert space factor of metric spaces

TL;DR

The paper proves that every complete metric space X decomposes uniquely as X = Y × H, where H is a (possibly finite- or zero-dimensional) Hilbert space and Y contains no splitting lines. It introduces the notion of factor subsets and a Zorn-lemma argument to construct a maximal Hilbert-factor H_x through any point x, then shows the complementary factor Y has no splitting lines and that H_x equals the union of all splitting lines through x. This yields a canonical isometry decomposition with Iso(X) ≅ Iso(Y) × Iso(H). The results connect to de Rham-type decompositions and extend to various geometric contexts, including CAT(0) and Alexandrov spaces, by characterizing the Hilbert-factor via splitting lines and projections. The method hinges on the interplay between factor subsets, their intersections, and Zorn’s Lemma to secure maximal Hilbert-factors.

Abstract

We prove that any complete metric space has a unique decomposition as a direct product of a possibly finite or zero-dimensional Hilbert space and a space that does not split off lines.

Paper Structure

This paper contains 8 sections, 4 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main observation
  4. Factor subsets
  5. Intersections and projections of factor subsets
  6. Splitting lines
  7. Zorn's Lemma
  8. End of Proof

Key Result

Theorem 1.1

Let $X$ be a complete metric space. Then there exists a unique decomposition of $X$ as a direct product $X=Y\times H$, where the metric space $Y$ does not contain splitting lines and $H$ is a (possibly finite- or zero-dimensional) Hilbert space. For any point $x\in X$, the maximal Hilbert space fact

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • proof : Proof of Theorem \ref{['thm: main']}