Strength of the Krein–Šmulian Theorem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Work over $\mathsf{RCA}_0$, the subsystem of second-order arithmetic with $\Delta^0_1$ comprehension and $\Sigma^0_1$ induction; all structures and orders are countable and coded by sets of integers. $\mathsf{ACA}_0$ adds comprehension for every arithmetical formula (set parameters allowed), and $\mathsf{ATR}_0$ adds arithmetical transfinite recursion along every coded well-order. Reverse-mathematical equivalence means provability of both implications over this base. Let $X$ be a separable real Banach space and $X^*$ its continuous dual. The weak-star topology $\sigma(X^*,X)$ is pointwise convergence on $X$. A set $C\subseteq X^*$ is bounded-weak-star closed if for every $r>0$, $C\cap\{f:\|f\|\le r\}$ is weak-star closed. The Krein--Šmulian theorem asserts that a convex $C$ is weak-star closed exactly when it is bounded-weak-star closed.

Statement recorded in the supplied source.

Determine the exact reverse-mathematical strength of this separable Krein--Šmulian theorem over $\mathsf{RCA}_0$. [S2]

Short English statement

Which axioms exactly suffice to infer weak-star closedness of a convex dual set from closedness on every norm ball?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500205 (AMR-074-0205). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stephen G. Simpson, Open Problems in Reverse Mathematics (15 December 1999), Banach Space Theory, second example. https://sgslogic.net/t20/talks/cta/problems/node3.html.