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27 INVARIANTS A N D INVARIANT MANIFOLDS for all xeX. This problem is posed in "general terms" but must be considered in the local theory format. It is found that it can be solved with the aid of the universal invariant. Theorem. Let a nonzero vector ηεΧ and a vector field ξ: V X, of the class C^(V), be given such that ξ(χ0) φ 0 for x0e V. 1) holds. Proof. 5 are used here. The universal invariant J of the group 1 ϋ (ξ) is the mapping X = Τ 0 Y Y and the equation J(r, y) = ζ can be solved uniquely for y, with any ζ taken from a sufficiently small neighborhood of the point z 0 = J(x0).

Which are x' = x, y' = y + a. 12 since the manifold y = 0 is not regularly defined by the 2 equation y = 0. §4. THE CONTINUATION THEORY 1. Spaces of Poly linear Mappings k Let X and Y be Banach spaces. The mapping φ: X -> Y is called polylinear (k-linear) if it is continuous at zero and if, for every /, 1 < / < /c, the equality φ ( χ ι , . . , + α'φ<χ axl + a'x\, x l +1 ? . , x f c) X/_ ! , X/, XJ+1, · · · , Xk) X/_ J, X Z , X/+ I, . . , is satisfied for any vectors x l 9 . . , xk9 x\ e X and any numbers a, a' e R.

3), rewritten as D(C,Ô) = (Ç-Ô)D + D É - D , 00 00 another commutator is calculated to be 2 2 [ D , ζ · δ] = Ζ ) ( ζ · < ? ) - ( £ · Ö)D 00 00 2 = D((C - 3)D + D £ . D ) - (ζ . D + ΙΩξΌ . 3) is a particular case (when k = 1). These calculated standard commutators will be used in Sections 19 and 24. oo 10. , 1 η" ). 1) ο where 3 f = dxi and dk = dvk. , ο k symmetric relative to permutations of the lower indices, each of which takes the values 1 , . . , n, while the index / takes the values 1 , .