Bounded variation and the strength of Helly's selection theorem

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 (BV ). For this we utilize a new representation of this space intermediate between L1 and the Sobolev space W , compatible with the—so called—weak∗ topology on BV . We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice. In this paper we investigate the space of functions of bounded variation (BV ) and Helly’s selection theorem (HST) from the viewpoint of reverse mathematics and computable analysis. Helly’s selection theorem is the most important compactness principle on BV . It is used in analysis and optimization, see for instance [1, 3]. This continues our work in [10] and [12] where (instances of) the Bolzano-Weierstraß principle and the Arzelà-Ascoli theorem were analyzed. There we showed, among others, that an instance of the Arzelà-Ascoli theorem is equivalent to a suitable single instance of the Bolzano-Weierstraß principle (for the unit interval [0, 1]), which, in turn, is equivalent to an instance of WKL for Σ1-trees. Here, we will show that an instance of Helly’s selection theorem is equivalent to a single instance of the Bolzano-Weierstraß principle (and with this to an instance of the other principles mentioned above). It is a priori not clear that this is possible since the proof of HST uses seemingly iterated application of the ArzelàAscoli theorem and since there are compactness principles, which are instance-wise strictly stronger than Bolzano-Weierstraß for [0, 1]. (For instance the Bolzano-Weierstraß principle for weak compactness on l2 has this property, see [11].) A fortori this shows that HST is equivalent to ACA0 over RCA0.