Department of Mathematical Sciences
Carnegie Mellon University
Orthogonal Projections and the Riesz Representation Theorem in Weak
Subsystems of Second Order Arithmetic
In the book Subsystems of Second Order Arithmetic by Stephen Simpson, the author states that the main question of Reverse Mathematics is Which set existence axioms are needed to prove the theorems of ordinary, non-set-theoretic mathematics? Here, ordinary means independent of abstract set-theoretic concepts, e.g. all the objects considered are countable or separable. It turns out that in many cases only a few such axioms are repeatedly used. The name reverse mathematics comes from the fact that, the reverse is also true - if certain axioms prove a theorem, they are logically equivalent to it. Everything takes place in second order arithmetic.
I am going to define the language of second order arithmetic, give axioms for the subsystems that I will be using (RCA0, WKL0, ACA0) and provide the necessary definitions. Next I am going to give conditions for the existence of orthogonal projections in Hilbert spaces. Then I will state the Riesz Representation Theorem and prove that it is equivalent to a number of other statements. The main result will be that both the existence of orthogonal projection onto closed subspaces of a Hilbert space and the Riesz Representation Theorem are equivalent to ACA0.