Every locally bounded space with trivial dual is the quotient of a rigid space

Abstract

Letting $ T_p$ denote the class of separable $p$-Banach spaces (for $0<p<1$) with trivial dual, we show that $ T_p$ does not have any projective spaces, \ie, there is no space $X$ in $ T_p$ such that every space in $ T_p$ is a quotient of $X$. In lieu of a projective space we construct the $L_p(w)$ spaces, which are structurally similar to the space $L_p$. We then define a particularly well behaved type of $L_p(w)$ space, namely the uniform $L_p(w)$ spaces, and we show that every space in $ T_p$ is a quotient of some uniform $L_p(w)$ space. We then define a badly behaved type of $L_p(w)$ space, namely the unbalanced biuniform L$_p$(w) spaces. If $ L_p(w)$ is unbalanced biuniform and $C$ denotes the one dimensional subspace of constant functions, then $ L_p(w)/C$ is a rigid space. We then show that each space in $ T_p$ is a quotient of one of these rigid spaces. This last result is used in an essential way to prove the nonexistence of a projective space in $ T_p$.