Search

10 Jan 2018: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

23 Feb 2018: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

9 Oct 2018: Show that f isbounded. 11. Show that there is an infinite set S of positive integers such that the sum of any two distinct elements. ... b) a collection of 100 horizontal lines and 100 vertical lines such that the 10,000 points of intersection are.

19 Jan 2018: Show that I Thas a square root, i.e., there exists S B(X) such that S2 = I T. ... 14+. Let X be a separable infinite-dimensional Banach space. Suppose thatthere is a constant C such that every finite-dimensional subspace E of X satisfiesd(E,n2 ) C, where

4 Feb 2018: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

8 Oct 2018: 1. Let E be a Lebesgue measurable subset of the real line with positive Lebesgue measure m(E).Show that for every > 0 there exists an open interval (a,b) such ... A Dirac mass at x X is themeasure δx on B such that δx(A) = 1 or 0 according as x A or x /

29 Oct 2018: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

9 Oct 2018: B. Prove that (A4B)4C = A4(B4C). 7. Let A1, A2, A3,. be sets such that A1 A2. ... c there is an x with a < x < b such that f(x) = c.

26 Nov 2018: 13. Let S PN be such that if A,B S then A B or B A. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A,B T?

12 Oct 2018: x). On the other hand, show that if fn does not convergeuniformly to f then we can find a convergent sequence xn x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)