Abstract
Suppose that κ is an infinite cardinal, Vκ=⊕α<κRα is a vector space of dimension κ over R, τκ is the box topology on Vκ, μκ and νκ are the maximal and maximal locally convex vector topologies on Vκ respectively. We prove that τω=μω=νω but, for every κ>ω, μκ⊃νκ⊃τκ. For every κ, the topological vector spaces (Vκ,μκ), (Vκ,νκ), (Vκ,τκ) are complete but not sequential.
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