We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic. In particular, we explore issues having to do with distances, closed subsets and subspaces, closures, bases, norms, and projections. We pay close attention to variations that arise when formalizing definitions and theorems, and study the relationships between them. For example, we show that a natural formalization of the mean ergodic theorem can be proved in ACA 0 ; but even recognizing the theorem's "equivalent" existence assertions as such can also require the full strength of ACA 0 .

The pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA0; moreover, the pointwise ergodic theorem is equivalent to (ACA) over the base theory RCA0.