Question 2: Simple proofs (10 marks) (a) Let S C C be a bounded set in the complex plane. Define the closure of S to be

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Question 2 Simple Proofs 10 Marks A Let S C C Be A Bounded Set In The Complex Plane Define The Closure Of S To Be 1 (27.75 KiB) Viewed 56 times

Question 2: Simple proofs (10 marks) (a) Let S C C be a bounded set in the complex plane. Define the closure of S to be the set S = SUOS. Show that (i) S is bounded: (ii) S is closed (b) Prove Corollary 13.3, i.e. the statement "The maximum modulus of a function that is holomorphic in a bounded domain, and contin- uous on the closed set Quan is always attained at the boundary." As part of this proof you are expected to show that (i) the function : C-R that sends 2 to 1z| is continuous. (You may use this to conclude that if is continuous wherever f is); (ii) the maximum is attained; (iii) the maximum is located at the boundary. You may use the outcome of part (a) as well as results stated on lecture slides or on the Problem Sheets.