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5 Dec 2019: compact. 5. Are the invertible operators dense in B(l2)? 6. Construct a self-adjoint operator T on a (non-zero) Hilbert space H such that T hasno eigenvalues. ... 7. Construct a compact operator T on a (non-zero) Hilbert space H such that T has

5 Feb 2019: Show that for any b C, there exists a sequence of pointszn 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.

25 Jan 2019: Can such a map be a Möbiustransformation? 10. Calculateγ z sin z dz when γ is the straight line joining 0 to i. ... functions of which theyare derivatives) on the domains indicated:. (a)1. z, (0 < |z| < ); (b) z.

27 Nov 2019: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

16 Oct 2019: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

16 Oct 2019: 7. Let 1 < p, q, r < be such that 1/p 1/q 1/r = 1. ... Must Y Z be dense in X? 9. Give an example of a vector space V and two inequivalent norms ‖ ‖A and ‖ ‖B onV such that the normed spaces (V,‖ ‖A)

6 Feb 2019: Show that for any b C, there exists a sequence of pointszn 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.

23 Jan 2019: But for such a fibre bundle we have. B. f(ch4kd(ξH) σ4kd) = 0,. ... Suppose that the fibration is in addition equipped with a map τ : E BOand a section s : B E, such that τ s : B BO is the constant map to thebasepoint.

14 Oct 2019: 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 /

27 Nov 2019: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

28 Feb 2019: 2. Let X and Y be N(0, 1) random variables such that cov(X,Y ) = 0. ... a random set C(Y ) such that Pβ,σ2 (b C(Y )) = 1 α for allβ,σ2.

12 Nov 2019: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

22 Oct 2019: to the zero function but with 10fn(x) dx 6 0. Is it possible to find such a sequence with. ... Onthe other hand, show that if (fn) does not converge uniformly to f, then there is a convergentsequence xn x in [a, b] such that fn(xn) 6 f(x).

5 Nov 2019: A = {x M : (xn) in A such that xn x} ,. Define the interior of A (in M) to be the set. ... Deduce that if A and B are disjoint closed subsets of M, then there exist disjoint opensubsets U and V of M such that A U and B V.

17 May 2019: Show. there exist disjoint open sets U1 and U2 such that Ci Ui.8. ... continuous path. f : [0, 1] Xconnecting a and b such that f is a homeomorphism onto its image.

6 Oct 2019: map(B,M) = {families of (.) over B}. Then. H(M) = {characteristic classes of such families}. ... This “resolution” point of view applies to many further examples,such as Confn(Rk) or BDi (W).

25 Nov 2019: everywhere.(b) If f is a measurable function on X such that f = f T almost everywhere, show that there is. ... Xn). Assume that F = σ(X1,X2,. ). Show that for every > 0, and every A F,there is n 1 and B Fn such that P(A4B) <. 3. 13. (Markov shift)

28 Jan 2019: Can such a map be a Möbiustransformation? 10. Calculateγ z sin z dz when γ is the straight line joining 0 to i. ... functions of which theyare derivatives) on the domains indicated:. (a)1. z, (0 < |z| < ); (b) z.

29 Nov 2019: V such that A Uand B V. ... I) Mn such that g(A)2 = A for all A Dr(I).

23 Apr 2019: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has