Question

Please do the problem 12.26 and 12.27.

Please do the problem 12.26 and 12.27.

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and aA < 1, but still assume that (12.1) 00 a) Show that T still maps B (yo) into itself. b) Show that T is a compact function. Exercises 12.25-12.28 require some knowledge of the theory of functions of a compla variable. In these exercises, 12 denotes an open subset of C and H(12) the set of fina tions that are analytic on 12. 12.25 Show that H(12) is a closed subset of C(2). Hint: Use the Cauchy integral formula. 12.26 Let FCH(12). Show that F is compact if and only if it is closed in C(12) and for each zo E N2 there is an r >0 such that Br(zo) CN and sup{\f(): feF, z EB(20)}<0. 12.27 Let 2 =B1(0) and U = {fe H(12): If <1}. Prove that for each z E2, there is a g EU such that g'(2)] = sup{\f'(): f EU}. 12.28 Let 12 and U be as in Exercise 12.27 and 0 <r <1. Consider the function T:U → U defined by T(f)(x) = f(rz). Prove that T is a compact function. ACTNESS OF PRODUCT SPACES The Cartesian product of two compact intervals of R is a compact rectangle in R?. Indeed, it is not hard to prove that I x A is a compact space whenever I and are comnact. spaces, as you are asked to show in Exercise 1990 1 m onolition of the foot

Answer 1

12.26 Let F CHCR) F is compact, Suppose claim: F is closed in C(2) & Bounded It is enough to show is closed in H(a) As Hees is closed in (Col) & bounded On ((a) we have folloning metric d(fig) & max ax 2hPa (fog) ditis HP ncf-9) Closed & Boundednen where Pa (f) = sup ( lfcan) atka! & don Uka kn an Compact knan Compact Inc katl So Hoss Also menisable Compactness in metric space - Compactness so we an done =

Conversy suppon F CHCR) is closed & Bounded in for each rood 7070 st. By (zo) ca. and Supſfee) : fer , 26 By (20) } co. - claim. f is compact As is Rounded - 4 Maco. Sit. By 1 f(2) / CM. & FEF & ZE Br(z.). Montel's classical theorum about normal families every segrence Efiz CF har subseg that converges uniformly on compact subset of R. Hence in H(-2)} to some fetch) Ar & is closed a FEF & is compact (Every stansese has convergent subsey) Sequential Compactness Compact in metricspace. 12.27) Let u = B, (o) #670, U-EFE H(2) | Iflg U is closed #fE U fn Eust Aalst & Kfn-fice = iflette Are is a bitan & FEU}. Also is bounded

proved in 12.26) By Heine Borel property U is compact. Consider fn EU Tfnicz) - sup { 1 fund 2) Il feos. Ar vis compact each sea has cgt subseq? so Ifaal- a gis required element