Schm\"udgen's theorem and results of positivity

This is a translation of a final paper. It contains proofs of some well-known Positivstellens"atze. The approach in this paper differs in two respects from presentations in literature. We use the Stone-Weierstrass Theorem directly to prove an inductive property, which makes the Kadison-Dubois Theorem superfluous in our paper, and we also consider homogeneous and projective Positivstellens"atze. This leads us to a homogeneous version of Schm"udgen’s theorem and a projective version of Putinar’s theorem.