Locally Complemented Subspaces and ℒ_p‐Spaces for 0 < p < 1

Abstract

Abstract We develop a theory of ℒ︁ p ‐spaces for 0 < p < 1, basing our definition on the concept of a locally complemented subspace of a quasi‐BANACH space. Among the topics we consider are the existence of basis in ℒ︁ p ‐spaces, and lifting and extension properties for operators. We also give a simple construction of uncountably many separable ℒ︁ p ‐spaces of the form ℒ︁ p ( X ) where X is not a ℒ︁ p ‐space. We also give some applications of our theory to the spaces H p , 0 < p < 1.