Singularities in quasi-Banach spaces with applications to semilinear elliptic equations .1.

The paper deals with the extension of the singularity theory which was elaborated for Banach spaces to quasi-Banach spaces. For general quasi-Banach spaces some striking tools fail: not locally convex, the dual space may be trivial,.... Applying the concept of dual rich quasi-Banach spaces and results about Fredholm maps some results of the classical theory can be carried over. As an application we show in the continuation of this part how one can use these results for the investigation of the solution structure of semilinear elliptic boundary value problems in function spaces of Besov-TriebeI-Lizorkin type.