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Group-norms are vector-space norms but with the scalars restricted to units (invertibles), ±1. The Birkhoff–Kakutani theorem (a first-countable Hausdorff topological group has a right-invariant metric) we view as a normability theorem rather than a metrization theorem, a relative of Kolmogorov’s normability theorem for topological vector spaces (the condition for whose normability is that the origin have a convex bounded neighbourhood). The groups here need not be abelian, so one has left-sided and right-sided versions. Proved here is the Analytic Baire Theorem: if a normed group contains an (either-sided) non-meagre analytic set, it is Baire, separable and (modulo a meagre set) itself analytic. Other results here include the ‘Analytic Shift Theorem’ and the ‘Squared Pettis Theorem’, category relatives of the Steinhaus Difference Theorem.

We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.

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6 - Groups and Norms: BirkhoffKakutani Theorem

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COMPUTABLE TOPOLOGICAL GROUPS

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6 - Groups and Norms: BirkhoffKakutani Theorem

Summary

COMPUTABLE TOPOLOGICAL GROUPS

6 - Groups and Norms: BirkhoffKakutani Theorem