Quotient groups of normed spaces for which the Bochner theorem fails completely (Q2762073)

A (Hausdorff) topological Abelian group \(G\) is said to be a B-group if it satisfies Bochner's theorem for locally compact Abelian groups: every positive definite function \(\phi\) on \(G\) with \(\phi(0)=1\) equals the (pre)-Fourier transform of a (necessarily unique) Radon measure on \(\Gamma\), the character group of \(G\), equipped with the compact open topology. On the other extreme \(G\) is said to be NBT (no Bochner theorem) if it admits a non-trivial positive definite function but \(\Gamma=\{ 0\}\). In p. 111 of [Additive subgroups of topological vector spaces, Lect. Notes Math., Vol. 1466 (Springer, Berlin, 1991; Zbl 0743.46002)] \textit{W. Banaszczyk} conjectured that every non-nuclear locally convex metrizable space \(E\) contains a closed subgroup \(K\) such that \(E/K\) is NBT. In this note the author advances the solution of this conjecture by proving that every infinite-dimensional real normed space contains a discrete subgroup \(K\) such that (span~\(K)/K\) is NBT.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].