Sequence spaces with small \(\beta\) -duals (Q1079777)

We attempt to make precise the following intuitive principle: ''A sequence space must be large if its \(\beta\)-dual is small''. This leads us to study FK-spaces E in which every subspace, with \(\beta\)-dual \(E^{\beta}\), is barrelled. The spaces \(c_ 0\), \(\ell^ p\), \(1<p<\infty\), cs and c enjoy this property, whereas \(\ell^ 1\), \(bv_ 0\), bv, bs, m and \(\omega\) do not. The key tool used here is a slight sharpening of the classical Silverman-Toeplitz theorem.