Bounded variation and the strength of Helly's selection theorem

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-Weierstraß principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.

References in corpus (4)

• The Bolzano-Weierstrass Theorem is the Jump of Weak König's Lemma
• Randomness and Differentiability
• Algorithmic randomness, reverse mathematics, and the dominated convergence theorem
• Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$

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