Locally Complemented Subspaces and ℒ︁p-Spaces for 0 < p < 1

We develop a theory of ??_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 ??_p-spaces, and lifting and extension properties for operators. We also give a simple construction of uncountably many separable ??_p-spaces of the form ??_p(X) where X is not a ??_p-space. We also give some applications of our theory to the spaces H_p, 0 < p < 1.