Computability of Banach Space Principles

We investigate the computable content of certain theorems which are sometimes called the “principles” of the theory of Banach spaces. Among these the main theorems are the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also study closely related theorems, as Banach’s Inverse Mapping Theorem, the Theorem on Condensation of Singularities and the BanachSteinhaus Theorem. From the computational point of view these theorems are interesting, since their classical proofs rely more or less on the Baire Category Theorem and therefore they count as “non-constructive”. These theorems have already been studied in Bishop’s constructive analysis but the picture that we can draw in computable analysis differs in several points. On the one hand, computable analysis is based on classical logic and thus can apply stronger principles to prove that certain operations are computable. On the other hand, classical logic enables us to prove “semi-constructive” versions of theorems, which can hardly be expressed in constructive analysis. For instance, the computable version of Banach’s Inverse Mapping Theorem states that the inverse T−1 of a linear computable and bijective operator T from a computable Banach space into a computable Banach space is computable too, whereas the mapping T 7→ T−1 itself is not computable. Thus, there is no general algorithmic procedure to transfer a program of T into a program of T−1, although a program for T−1 always exists. In this way we can explore the border between computability and non-computability in the theory of Banach spaces. As applications we briefly discuss the effective solvability of the initial value problem of ordinary linear differential equations and we prove the existence of computable functions with divergent Fourier series. The focus of our investigation is mainly on infinite-dimensional separable Banach spaces, but also the finitedimensional case, as well as the non-separable case, will be discussed.