Introduction

This paper is part of an ongoing study of the role of set existence axioms in the foundations of mathematics. The ongoing study has been carried out in the context of subsystems of second order arithmetic, under the slogan Reverse Mathematics [12,3,23]. We continue this program here by examining the role of strong set existence axioms in separable Banach space theory. We show that a very strong set existence axiom is needed in order to prove basic results concerning the weak-* topology on the dual of a separable Banach space.

The results in this paper are related to earlier work of Brown and Simpson [5,7,6,23] and Shioji and Tanaka [22]. The earlier work shows that the basic notions of separable Banach space theory can be developed in very weak subsystems of second order arithmetic, and that many basic results can be proved in such systems. Specifically, the Hahn-Banach theorem and a version of the Schauder fixed point theorem are provable in $\mathsf{WKL}_0$; the Banach-Steinhaus theorem is provable in $\mathsf{RCA}_0$; and versions of the Open Mapping and Closed Graph theorems are provable in $\mathsf{RCA}_0^+$. The set existence axioms of these three subsystems of second order arithmetic are very weak, in the sense that the systems themselves are conservative over Primitive Recursive Arithmetic for $\Pi^0_2$sentences (see Chapter IX of [23]). In particular, the mentioned systems are considerably weaker than first order arithmetic. Thus the results of Brown and Simpson [5,7,6] may have tended to support the opinion that only very weak set existence axioms are needed for separable Banach space theory. Our main results here, Theorems 5.6 and 5.7 below, provide a counterexample to that opinion and a departure from Brown-Simpson-Shioji-Tanaka. Namely, Theorems 5.6 and 5.7 show that a very strong set existence axiom, $\Pi^1_1$ comprehension, is needed in order to prove basic facts such as the existence of the weak-* closure of any norm-closed subspace of $\ell_1=c_0^*$. Thus $\Pi^1_1$ comprehension is in a sense indispensable for separable Banach space theory. This is significant because $\Pi^1_1$ comprehension is, of course, much stronger than first order arithmetic.

As a byproduct, we show that the Krein-Smulian theorem for the dual of a separable Banach space (Theorem 2.7 below) is provable in $\mathsf{ACA}_0$ (Theorem 4.14 below). We conjecture that the Krein-Smulian theorem for the dual of a separable Banach space is actually provable in the weaker system $\mathsf{WKL}_0$.

Some of our results here may be of interest to readers who are familiar with Banach spaces but do not share our concern with Reverse Mathematics and other foundational issues. Namely, the following Banach space phenomenon may be of independent interest. Let Z be a subspace of the dual of a separable Banach space. Banach and Mazurkiewicz observed that, although the weak-* closure of Z is the same as the weak-* sequential closure of Z, it is not necessarily the case that every point of the weak-* closure of Zis the weak-* limit of a sequence of points of Z. Indeed, the process of taking weak-* limits of sequences may need to be iterated transfinitely many times in order to obtain the weak-* closure. In a self-contained part of this paper, we obtain a sharp result along these lines. Namely, for each countable ordinal $\alpha$, we obtain an explicit example of a norm-closed, weak-* dense subspace of $\ell_1=c_0^*$ whose weak-* sequential closure ordinal is exactly $\alpha+1$. This result is originally due to McGehee [18], but our examples are different and more elementary.

On the other hand, it is perhaps worth noting that our original motivation for the work here had nothing to do with Banach space theory. Rather, our starting point was another aspect of Reverse Mathematics, specifically the search for necessary uses of strong set existence axioms in classical (``hard'') analysis. We began with the thought that, in searching for necessary uses of strong set existence axioms, it would be natural to consider how Cantor was led to the invention or discovery of set theory in the first place. We were struck by the well known historical fact [9,10] that Cantor introduced ordinal numbers in tandem with his study of trigonometric series and the structure of sets of uniqueness; see also Jourdain's essay [8]. Indeed, Cantor's proof that every countable closed set is a set of uniqueness uses transfinite induction on the Cantor-Bendixson rank of such sets. Therefore, from our Reverse Mathematics viewpoint, it is very natural to reexamine these results of Cantor. Although we postpone such reexamination to a future paper, we want to point out that our work here was inspired by a discussion of Kechris and Louveau [14,13] culminating in a result attributed to Solovay: the Piatetski-Shapiro rank is a $\Pi^1_1$-rank on the set of closed sets of uniqueness. Since the Piatetski-Shapiro rank is the weak-* sequential closure ordinal of a certain weak-*dense subspace of $\ell _1$, our foundational motivation for studying such ordinals is apparent.

We end this introductory section with a brief outline of the rest of the paper. Section 2 reviews the concepts and results of Banach space theory that are important for us here. In particular we review the weak-* topology and define the notion of the weak-*sequential closure ordinal of an arbitrary set in the dual of a separable Banach space. In Section 3 we exhibit the previously mentioned examples concerning weak-* sequential closure ordinals, using the concept of a smooth tree. These two sections, Sections 2 and 3, are intended to form a self-contained unit which should be accessible to anyone who is familiar with the notion of a Banach space. Our discussion of subsystems of second order arithmetic does not get under way until Section 4. We begin that section by reviewing the definitions and results from Brown-Simpson [5,7,6,23] that we shall need. We then discuss the weak-* topology and related notions in the Brown-Simpson context. We end Section 4 by proving our version of the Krein-Smulian theorem within $\mathsf{ACA}_0$. Finally, in Section 5, we state and prove our main theorem, concerning the need for $\Pi^1_1$ comprehension. The ideas of Section 3 are used in the proof of the main theorem in Section 5.