A characterization of monotone continuous linear functionals on tensoralgebras which arise in QFT is derived and some consequences are investigated. Then we look for necessary and sufficient conditions on a set $$T_{(N)} = \{ 1,T_1 ,T_2 ,...,T_N \} T_n \in E'_n $$ of “n-point-functionals”, which guarantee the existence of at least one monotone continuous linear functional $$S = \{ 1,S_1 ,S_2 ,...\} {\text{ }}on{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{E} = \mathop \oplus \limits_{n = 0}^\infty {\text{ }}E_n ,{\text{ }}E_n = E_1 \tilde \otimes _\pi E_1 \tilde \otimes _\pi ...\tilde \otimes _\pi E_1 ,$$ E 1 a special nuclear space, such that\(S \upharpoonright \mathop \oplus \limits_{n = 0}^N {\text{ }}E_n = T_{(N)} \), with special attention to QFT. A first application is a characterization of all monotone continuous linear extensions in the caseN=2. The notion of minimal extensions is introduced. Its relevance is discussed. Necessary and sufficient conditions onT (2N) for the existence of minimal extensions are presented. Some properties of minimal extensions are derived. In the simplest caseE≅ℂ the concept of minimal extensions allows to answer the extension problem completely for arbitraryN∈ℕ. For the case of generalE=E 1 andN=2 it is shown that the known examples of monotone continuous linear extensions are minimal extensions or a special generalization of it.