Some Examples of Non-Metrizable Spaces Allowing a Simple Type-2 Complexity Theory

Representations of spaces are the key device in Type-2 Theory of Effectivity (TTE) for defining computability on non-countable spaces. Almost-compact representations permit a simple measurement of the time complexity of functions using discrete parameters, namely the desired output precision together with “size” information about the argument, rather than continuous ones. We present some interesting examples of non-metrizable topological vector spaces that have almost-compact admissible representations, including spaces of real polynomial functions and of distributions with compact support.