Which set existence axioms are needed to prove the separable Hahn-Banach theorem?

Abstract

We work in the context of weak subsystems of second order arithmetic. RCA 0 is the system with Δ 1 0 comprehension and Σ 1 0 induction on the natural numbers. WKL 0 is RCA 0 plus weak König's lemma for trees of finite sequences of 0's and 1's. Within RCA 0 we encode a separable Banach space  as a countable normed space A over Q . Points of  are Cauchy sequences from A which converge at the rate of at least 2 − n . We show that the Hahn-Banach theorem for separable Banach spaces is provably equivalent to WKL 0 over RCA 0. Thus, once again, WKL 0 is revealed as mathematically powerful, despite being proof theoretically equivalent to primitive recursive arithmetic.