Locally bounded noncontinuous linear forms on strong duals of nondistinguished Köthe echelon spaces

Abstract:

In this note it is proved that if ${\lambda _1}(A)$ is any nondistinguished Köthe echelon space of order one and ${K_\infty } \simeq {({\lambda _1}(A))’_b}$ is its strong dual, then there is even a linear form :${K_\infty } \to {\mathbf {C}}$ which is locally bounded (i.e. bounded on the bounded sets) but not continuous. It is also shown that every nondistinguished Köthe echelon space contains a sectional subspace with a particular structure.