Analytical properties of resource-bounded real functionals

Computable analysis is an extension of classical discrete computability by enhancing the normal Turing machine model. It investigates mathematical analysis from the computability perspective. Though it is well developed on the computability level, it is still under developed on the complexity perspective, that is, when bounding the available computational resources. Recently Kawamura and Cook developed a framework to define the computational complexity of operators arising in analysis. Our goal is to understand the effects of complexity restrictions on the analytical properties of the operator. We focus on the case of norms over C[0,1] and introduce the notion of dependence of a norm on a point and relate it to the query complexity of the norm. We show that the dependence of almost every point is of the order of the query complexity of the norm. A norm with small complexity depends on a few points but, as compensation, highly depends on them. We briefly show how to obtain similar results for non-deterministic time complexity. We characterize the functionals that are computable using one oracle call only and discuss the uniformity of that characterization. This paper is a significant revision and expansion of an earlier conference version.