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26 Nov 2008: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12 e(G).

28 Feb 2008: In this and the following questions, youare not being asked to find such an isomorphism explicitly.)Decompose V 23 , Λ. ... b) Explicitly construct such a representation ρ : SU (2) SO(3) by showing that SU (2)acts on the vector space of matrices of the

4 Feb 2008: Let P be the set of all p Q such that 2 < p2 < 3. ... c) If A and B are closed subsets of R such that 1 A, 1 B and [1, 1] A B, thenA B 6=.

10 Mar 2008: Prove that the set S of points x [0, 1] such that x Aj for infinitely many j is dense.Must S be open? ... b) If f : (0, 1) R is continuous and bounded, prove that there exists a sequence of polynomialsQn such that Qn are uniformly bounded and Qn f

10 Mar 2008: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

17 Oct 2008: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 f

28 Feb 2008: n}, let the non-negative numbers A(n)k and the“nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

23 Feb 2008: 4) Let X be a space, x X, f : S1 X a map such that x is inthe image of f, and Y = X f B. ... 1) Showthat there is an element z C such that γ(z) and x′ in C′.

17 Nov 2008: In this and the following questions, you arenot being asked to find such an isomorphism explicitly.). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a

13 Nov 2008: a linear func-tional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspace exists,show that there exists a basis v, α(v),.