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By Prof. Dr. Gennadi M. Henkin, Prof. Dr. Jürgen Leiterer (auth.)

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Q-convex functions 4. 1. The Levi form. If M is a real c1 manifold and xeM, then by Tm,x

Setting and if r=n, we complete the proof. 18. 15. First consider the case r=O. Since X is compact, V(X,E) is a Banach space with the norm U·Uo,x· Since, by Montel's theorem, any bounded set in V(X,E) is relatively compact, this space is finitely dimensional. 13 (regularity of a), z00 , 0 = o. Now let 1~r~n. 17, we can find bounded linear operators and such that ( 1. 18) where id is the identical operator. Since, oy Ascoli's theorem, the embedding 33 is compact, Kr is compact as an operator from zg,r(X,E) into itself.

0. Furthet· on, for every collection K=(k 1 , ... ,k 1 ) of integers, -write IKI:=l and K(s~):=(k 1 , ... ,ks_ 1 ,ks+ 1 , ... ,k1 ) (1

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