On properties of subspaces of $l\sb{p},\,0<p<1$

Abstract

The material presented in this paper deals with some questions con- cerning projections, quotient spaces, and linear dimension in lp spaces, and also includes a remark about weak Schauder bases in lp spaces and an example of an infinite-dimensional closed subspace of /p which is not isomorphic to lp. A great deal is known about the structure of lp spaces for p 2; 1. Perhaps not quite so much is known about these spaces for 0<p< 1. We plan to discuss here proper- ties of subspaces of the latter spaces and show that in most cases these properties are quite different from those of the normed spaces. The paper will be divided into five sections. §1 will contain some comments and definitions which might be helpful to the reader as well as a summary of results. §2 will deal mainly with the concept of complemented subspaces of lp, 0<p<l. We will show in this section that each lp space is isomorphic to all of its subspaces of finite codimension, that each lp space contains a subspace isometrically iso- morphic to lp no infinite-dimensional subspace of which is complemented in /, and that if lp is isometrically isomorphic to one of its subspaces which has the Hahn-Banach extension property, then this subspace is complemented in lp. §3 will contain an example of a subspace of lp which is not isomorphic to lp. §4 will contain some results about subspaces of lp which are obtained as kernels of linear mappings. In particular, we will show that each lp, 0</;<l, contains a closed proper subspace such that any continuous linear functional on lp which vanishes on this subspace vanishes on all of lp. We will also show that lp contains a closed subspace which is not contained in any proper complemented subspace of lp. Finally, §5 will contain some results on linear dimension, complementing those known for lp,p^l. Most of our terminology is standard ; however, a few remarks probably are in order. We use the word norm to denote the lp paranorm when 0<p< I, and we