 
Summary: Fundamental notions of analysis
in subsystems of secondorder arithmetic
Jeremy Avigad
Department of Philosophy
Carnegie Mellon University
Pittsburgh, PA 15213
Ksenija Simic
Department of Mathematics
The University of Arizona
Tucson, AZ 85721
Abstract
We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces
in the context of subsystems of secondorder 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 for
malizing 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 ACA0; but even recognizing the theorem's "equivalent" existence asser
tions as such can also require the full strength of ACA0.
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