# Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

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DOI: 10.1007/s11854-008-0011-y

- Cite this article as:
- Schul, R. J Anal Math (2007) 103: 331. doi:10.1007/s11854-008-0011-y

## Abstract

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in ℝ^{d}. Their results formed the basis of quantitative rectifiability in ℝ^{d}. We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales aroundmost points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analogue of the Square function. This allows us to conclude that for a given set *K*, the ℓ_{2} norm of this quantity (which is a function of *K*) has size comparable to a shortest (Hausdorff length) connected set containing *K*. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90, Oki92] are **independent of the ambient dimension**.