By Gawronski W., Shawyer B. L., Trautner R.

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Additional resources for A Banach space version of Okada's theorem on summability of power series

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Let y and r1 be two closed paths in the plane C. If r never takes the value o and if we always have Ir 1(t) I < fr(t) [, then the mapping t - r(t) + r1(t) never takes the value o and I(r + r1, o) == I(r, o). For, we can write the closed path t - 1 + r11g] has zero index with respect to the origin because it is contained in the open disc of centre the closed path r + r1 COMPLEMENTS: and radius 8. 2, we obtain proposition ORIENTED BOUNDARY OF A 1. 3. is the product of two closed paths and by applying proposition 9· 1 1 Thus + r1 , r COMPACT SET LEMMA.

3. Any closed differential form has a primitive in D. 1 For, from theorem 2 , we have J in D, which implies by proposition w = 2. 1 w in a simply connected open set o for any closed path y contained that w has a primitive in D. In particular, in any simply connected open set not containing closed form D dz/z has a primitive; in other words, simply connected open set which does not contain o. log o, the z has a branch in any Examples of simply connected open sets. We say that a subset E of the plane is staffed with respect to one of its points a if, for any point z e E, the line segment joining a to z lies in E.

Such a primitive always exists and is unique up to addition of a constant, This lemma is, in some sense, an extension of theorem 1. it in an similar way. We shall prove By using the compactness of the rectangle, we can quadrisect it by subdividing the interval of variation of and that of u by points t by points t; uh in such a way that, for all i,j, the small rectangle, which is the product of the segments [ti> t1 + 1], [uh ui+1J, is mapped by o into an open disc U1,i> in which w has a primitive F1,i.

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