  This is my diploma thesis in german language. In the context of formal
deformation theorie of assoziative observables in classical field theory I
consider the symmetric algebra S(V) on an arbitrary-dimensional R- or
C-vectorspace V as a prototype of comprehensive observables algebras in quantum
field theory. In this framework I calculate the Hochschild cohomologies of S(V)
with values in S(V)-S(V)-bimodules M. In the case that V is a locally convex
vectorspace I compute the continuous Hochschild cohomologies for the (with help
of the pi-tensor product) locally convex topologised symmetric Algebra on V and
likewise for the completion Hol(V) of S(V) if V is in addition a Hausdorff
space and M is complete. For all this cases and in the situation of symmetric
bimodules M I prove generalized Hochschild-Kostant-Rosenberg theorems by use of
explicite chain maps. Furthermore I have found useful statements about the
differential Hochschild cohomologies in the case that M is a bimodule whose
right modul multiplication can be written as a sum of the left modul
multiplication and higher differential terms.
