L on(Q)' Uno on(Q) E n=O This is well-defined: actually we are dealing with finite sums only! We claim that (*) V(Q) + V(eT) C V(Q + eT) holds for all Q, eT EM. The assertion is trivial if Q,eTEIN o . Suppose now, for some NEIN o, that (*) holds for all Q,eTEM such that On(Q) = On(eT) = 0 for n>N.

Spaces of differentiable functions Let Q c: IR' be a (non-empty) open set, rEIN. , ... , s,) E IR', let us write I s I := i~l I Si land operator 11 s 11 := oSI+· .. +Sr aqt ... at:, (~1 s; Y2. If SEIN b, then we denote by as the differential . ;;; k is a subspace of CC(Q). By means of the F-seminorms P K,k (f) := max max I asj(x) Isl <;k I. XEK where K runs through the compact sub sets of Q, CC(k) (Q) be comes a tvs which is obviously Hausdorff. Since Q is locally compact and countable at infinity there is a sequence (Kn ) ofcompact subsets of Q such thatKn c: Kn + 1 , 'Vn EIN, and every compact sub set of Q is contained in some K n.