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The questions for the university of cambridge mathematical tripos part iii analysis paper, held on thursday, 2nd june, 2011, from 1:30 pm to 4:30 pm. The paper consists of four questions, covering topics such as continuous functions of two variables, complete metric spaces, convex sets, and distributions.
Typology: Exams
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Thursday, 2 June, 2011 1:30 pm to 4:30 pm
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
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Show that any continuous function of two variables f : [0, 1]^2 → R can be written in terms of continuous functions of one variable and addition.
Show that the following two statements are equivalent. (A) If (E, d) is a complete metric space, then E cannot be written as the union of a countable collection of closed sets with empty interior.
(B) Suppose that X and R are sets with R ⊆ X × X and such that
{y ∈ X : (x, y) ∈ R} 6 = ∅
for all x ∈ X. Then there exists a function G : Z++^ → X such that
G(n), G(n + 1)
for all n > 1.
Define the terms convex set, closed convex hull and extreme point as used in the statement that every compact convex set in a normed space over R is the convex hull of its extreme points. Prove the statement. State and prove Carath´eodory’s lemma about convex sums in Rn. Consider the space l^1 (R) of real sequences x with
j=1 |xj^ |^ convergent and norm ‖x‖ 1 =
j=1 |xj^ |. Find the extreme points of the closed unit ball Σ and show that not every point of Σ is a finite convex combination of extreme points. Show that if K is a compact convex set in Rn^ whose extreme points form a closed set, then every point in K is in the convex hull of n + 1 of its extreme points. Show by means of an example that we cannot replace n + 1 by n. If K is a compact convex set in Rn, is it true that K is the convex hull of finitely many extreme points? Give a proof or counterexample.
Part III, Paper 7
