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LANCASTER UNIVERSITY
2011 EXAMINATIONS
PART II (Final Year) MATHEMATICS & STATISTICS Math 317 Hilbert Space 2 hours
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. K denotes one of the number fields R or C. Recall the following result from the course.
Theorem Let U be a closed subspace of a Hilbert Space H.
(a) For each x ∈ H, there is a unique vector xU ∈ U such that ‖x − xU ‖ = dist(x, U ).
(b) Let x ∈ H. For u ∈ U , the following are equivalent: (i) (x − u) ∈ U ⊥; (ii) u = xU.
please turn over
SECTION A
A1. (a) What is meant by a norm on a vector space V over K? [5] (b) Let ^1 denote the set of all sequences z = (zn)n≥ 1 , where zn ∈ C for each n ∈ N, such that ‖z‖ 1 := ∑∞ n=1 |zn| < ∞. Show that ^1 is a vector space and ‖ · ‖ 1 is a norm on it. [HINT: You may assume that the set of all complex sequences z = (zn), forms a vector space over C.] [10]
A2. (a) State the Cauchy–Schwarz inequality for an inner product 〈 , 〉 on a complex vector space, giving the necessary and sufficient conditions for equality to hold. [5] (b) Show that (^) ∫ (^) π 0 3 t√sin t dt ≤ π√ 6 π. [8]
(c) Prove that the inequality in (b) is strict. [5]
A3. Let U be a closed subspace of a Hilbert Space H. (a) What is meant by U ⊥, the orthogonal complement of U? [3] (b) Show that U ⊥^ is a subspace of H. [4] (c) For x ∈ H, what is meant by the distance of x from U , dist(x, U )? [3] (d) Prove that (i) implies (ii), in the theorem recalled on the cover page. [7]
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SECTION B continued
B3. Let H and K be Hilbert spaces. (a) What is meant by saying that an operator T from H to K is bounded? [4] (b) (i) Let v ∈ H and let ϕ : H → C be the linear functional given by ϕ(x) = 〈v, x〉. Show that ϕ is bounded and ‖ϕ‖ = ‖v‖. [8] (ii) State the Riesz–Fr´echet Theorem. [4] (iii) Prove that the linear functional ϕ on ^2 given by
ϕ(z) =
∑^ ∞
n=
2 −nzn
is bounded, and find its norm. [6] (c) Let T be the operator on L^2 [0, 1] given by (T f )(x) =
∫ (^) x 0 f (t) dt. (i) Show that T is bounded. [4] [HINT: Express (T f )(x) in the form 〈gx, f 〉 for some function gx(t), and apply the Cauchy–Schwarz inequality.] (ii) Show that √^1 3 ≤ ‖T^ ‖ ≤^
√^1
2.^
[4]
[HINT: Apply T to the constant function equal to 1.]
end of exam
