On minimally subspace-comparable F-spaces

Authors:

• Lech Drewnowski

Abstract

An F-space (complete metric linear space) is minimal if it admits no strictly weaker linear Hausdorff topology, and quotient (q-) minimal if all of its Hausdorff quotients are minimal. Two F-spaces are (q-minimally) minimally s-comparable if they have no isomorphic (q-) nonminimal closed linear subspaces. It is proved that if X, Y are (q-minimally (resp., minimally) s-comparable F-subspaces of an arbitrary topological linear space E (resp., with X ∩ Y = {0}), then X + Y is an F-subspace of E. Also, if X₁,..., X_n are F-subspaces of E, then X₁ + ··· + X_n is an F-subspace of E, provided that X_i F and X_j G are minimally s-comparable whenever F and G are closed minimal subspaces of X_i and X_j, i ≠ j. These are analogs of some results due to Gurariǐ and Rosenthal concerning totally incomparable Banach spaces. © 1977.