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This textual content is a part of the Walter Rudin scholar sequence in complex arithmetic.

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Additional resources for Affine Functions on Compact Convex Sets (unpublished notes)

Example text

By compactness, a finite number of these open sets cover the unit sphere in H. , fm l . Now let linear functionals be m-dimensional space with the by Vh = (f i (h), f' 13 be the -norm. Define a map V : , fm (h)) , so that II hil II (1 - E ) II hi' for all h H. e. 1:4 VII %:1 - E , so in particular V is (1,1). Let V -1 denote the inverse of V mapping V(H) onto H with v- 1 11‘ (1 mapping 1: E ) -1 . 7 V-1 has an extension T into A(S) with II TIK(1 + )/( 1 - ). On the other hand by the Hahn-Banach theorem there, is a norm preserving extension U of V taking A(S) into i:.

By density of the -{ gni - gn 2 there are i, n with gi (k)

If x E C define Rx = If Rx 0 -1 Sxnf (Tx + U). c. and affine. That R is affine is easily checked. Let 0 V be open in E, and let W = open in E X F. c. c. is clear -30from the equality: Exec Rxn V  0/ = fxe C : (S x T)(x) nWn (VxF) Oi Before proving the main selection theorems, we show the existence of "approximate selections". 7. Let S be a compact simplex and E a locally convex space. c. and affine. If U is a convex neighbourhood of the origin in E, then let (i + U)(k) = Ii ( k) + U for all k ES.