The Tikhonov and Alaoglu–Bourbaki Theorems

Abstract

The central result of this chapter is the Alaoglu–Bourbaki theorem: Polars of neighbourhoods of zero in a locally convex space E are σ(E′, E)-compact subsets of E′. As a consequence in a dual pair 〈E, F〉 one concludes that, for a locally convex topology τ on E with (E, τ)′ = F, one always has σ(E, F) ⊆ τ ⊆ μ(E, F), where μ(E, F) is the Mackey topology on E, corresponding to the collection of absolutely convex σ(F, E)-compact subsets of F. As a prerequisite we show Tikhonov’s theorem, and as a prerequisite to the proof of Tikhonov’s theorem we introduce filters describing convergence and continuity of mappings in topological spaces.