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The third paper of the mathematical tripos part iii exam held on june 2, 2005. The paper focuses on functional analysis and banach algebras. Topics include the baire category theorem, continuous functions, and the resolvent of an element in a banach algebra. Students are required to attempt three questions out of four, which carry equal weight. The document also includes instructions for stationery requirements and special requirements for the exam.
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Thursday 2 June, 2005 9 to 12
Attempt THREE questions.
There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury tag Script paper
1 (i) State and prove the Baire category theorem.
(ii) By using the Baire category theorem, or otherwise, show that we can find an x ∈ Rn^ such that (^) n ∑
j=
kj xj 6 = kn+
whenever k 1 , k 2 ,... , kn+1 are integers, not all of which are zero.
(iii) Show that, given any K > 0 we can find a continuous function f : T → R such that ‖f ‖∞ 6 1 but the N -th partial Fourier sum SN (f, 0) satisfies
|SN (f, 0)| > K
for some N.
(iv) Show that there exists a continuous function whose Fourier series diverges at
Paper 6
3 Let B be a commutative Banach algebra with a unit. Develop the theory of the resolvent of an element x ∈ B up to and including the formula
ρ(x) = sup{|λ| : λe − x is not invertible}
for the spectral radius.
Give an example of a B and an x ∈ B for which ρ(x) = 0 although x 6 = 0. Give an example of a B and an x ∈ B for which ρ(x) = ‖x‖B = 1.
[You may assume results from the theory of vector valued integration but not from the theory of Banach algebra valued analytic functions.]
4 Let C([− 1 , 1]) be the space of real valued continuous functions on [− 1 , 1] under the uniform norm. Consider the subspace Pn of real polynomials of degree at most n. You may assume that, if T is a linear map Pn → R, with ‖T ‖ = 1, then, given > 0, we can
find an N > 1 and λ 1 , λ 2 ,... λN ∈ R with
j=1 |λj^ |^ = 1 and^ x^1 , x^2 ,... xN^ ∈^ [−^1 ,^ 1] such that (^) ∣ ∣ ∣ ∣∣T P^ −
j=
λj P (xj )
Show, proving the results (such as Caratheory’s theorem) that you need, that, if P is a real polynomial of degree at most n and u /∈ [− 1 , 1], then
|P (u)| 6 sup x∈[− 1 ,1]
|P (x)||Tn(u)|
where Tn is the Tchebychev polynomial of degree n.
Paper 6
