Computing with Sequences, Weak Topologies and the Axiom of Choice

We study computability on sequence spaces, as they are used in functional analysis. It is known that non-separable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice non-separable normed spaces cannot even be admissibly represented with respect to any compatible topology (a compatible topology is one which makes all bounded linear functionals continuous). Surprisingly, it turns out that when one replaces the Axiom of Choice by the Axiom of Dependent Choice and the Baire Property, then some non-separable normed spaces can be represented admissibly on Turing machines with respect to the weak topology (which is just the weakest compatible topology). Thus the ability to adequately handle sequence spaces on Turing machines sensitively relies on the underlying axiomatic setting.