Every locally bounded space with trivial dual is the quotient of a rigid space (Q1347356)

Let \(0<p<1\). The article deals with the class \(T_p\) of separable \(p\)-Banach spaces with trivial dual and the question about projective elements in this class. Utilizing increasing sequences of finite dimensional spaces with \(p\)-norms and suitable weight functions \(w\), the author exhibits an ingenious method of constructing a special subclass of \(T_p\), the elements of which are function spaces and are called \(L_p(w)\)-spaces. Those \(L_p(w)\)-spaces, which are in a natural sense uniform by construction, are called uniform \(L_p(w)\)-spaces. The author proves that the uniform \(L_p(w)\)-spaces are a projective subclass of \(T_p\) (i.e., each \(X\in T_p\) is a quotient of a suitable uniform \(L_p(w)\)-space). Next the author constructs another subclass of the class of \(L_p(w)\)-spaces, the so-called unbalanced biuniform \(L_p(w)\)-spaces, which also form a projective subclass of \(T_p\). More precisely: being a function space, each \(L_p(w)\)-space \(X\) contains the one-dimensional subspace \(C\) of constant functions. Then the class of quotient spaces \(X/C\), \(X\) an unbalanced biuniform \(L_p(w)\)-space, is projective in \(T_p\), and all such quotients \(X/C\) are rigid (i.e., admit only trivial endomorphisms). From these facts the author derives the result that \(T_p\) contains no projective elements.