Bounded variation and the strength of Helly's selection theorem

TL;DR

This paper investigates the logical strength of Helly's selection theorem for functions of bounded variation, establishing its equivalence to key principles in reverse mathematics and the Weihrauch lattice.

Contribution

It introduces a new intermediate representation of the BV space and characterizes HST's strength as equivalent to the Bolzano-Weierstrass principle and ACA0 in reverse mathematics.

Findings

HST is instance-wise equivalent to the Bolzano-Weierstrass principle over RCA0.

HST is equivalent to ACA0 over RCA0.

A similar classification is achieved within the Weihrauch lattice.

Abstract

We analyze the strength of Helly's selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. We obtain that HST is instance-wise equivalent to the Bolzano-Weierstra\ss\ principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.