On function spaces and polynomial-time computability

In Computable Analysis, elements of uncountable spaces, such as the real line R, are represented by functions on strings and fed to Turing machines as oracles; or equivalently, they are represented by infinite strings and written on the tapes of Turing machines [Wei00, BHW08]. To obtain reasonable notions of computability and complexity, it is hence important to choose the “right” representation (encoding) for the spaces being considered. Let’s say we have already agreed upon representations and ı of spaces X and Y (that are admissible with the topologies of X and Y ). How would we represent the space CŒX !Y  of continuous functions from X to Y ? It is known that there is a natural representation Œ ! ı of CŒX ! Y  which is characterized by the property that it is the poorest representation that makes function evaluation computable [Wei00, Lemma 3.3.14]. Is there a representation with a similar property also at the level of polynomial-time computability (as introduced in [KF82] and extended in [KC96, KC12])? In this note we observe that there is such a nice representation for the space of continuous real-valued functions. Generalization to other spaces is left for future research.