Bounded time computation on metric spaces and Banach spaces

We extend Kawamura and Cook's framework for computational complexity for operators in analysis. This model is based on second-order complexity theory for functionals on the Baire space, which is lifted to metric spaces via representations. Time is measured in the length of the input encodings and the output precision. We propose the notions of complete and regular representations. Completeness is proven to ensure that any computable function has a time bound. Regularity relaxes Kawamura and Cook's notion of a second-order representation, while still guaranteeing fast computability of the length of encodings. We apply these notions to investigate relationships between metric properties of a space and existence of representations that render the metric bounded-time computable. We show that time bounds for the metric can straightforwardly be translated into size bounds of compact subsets of the space. Conversely, for compact spaces and for Banach spaces we construct admissible complete regular representations admitting fast computation of the metric and short encodings. Here it is necessary to trade time bounds off against length of encodings.