On a family of topologies on the spaceL _p

Abstract

We study a family of topologies {ϰ_s}_0<s≤p on the space l_p, 0<p<∞. Here ϰ_s is the protective topology on l_p generated by the family of multipliers m_y:l_p→l_s, m_y(x)=x · y, where y ranges over the space l_p and 1/p + 1/q=1/s. Here l_s is taken with its standard topology generated by the norm for s ≥1 or a pseudonorm if 0<s< 1. It is shown that the family {ϰ_s}_{0≤s≤p is strictly increasing and that all the topologies ϰs}, 0<s< p, generate the same convergence of sequences. It is also shown that the topologies ϰ_s are not locally convex when 0<s≤min(1,p).