Abstract
This paper deals with the computability in analysis within the framework of Grzegorczyk's hierarchy, which is in the number 1 of addendum of open problems in Pour-El and Richards ([5], Computability in Analysis and Physics, Springer, Berlin, 1989). We combine two concepts, computability for sequences of real-valued functions and Grzegorczyk's hierarchy for recursive number theoretic functions, together and examine the computability in analysis restricted to primitive recursion and below. The notions of ( E r ) primitive computability structures on Banach space, in particular, for sequences of reals and real-valued functions are introduced; relations between ( E r ) primitive computability structures are proved; some basic properties are studied.
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