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LANCASTER UNIVERSITY
2010 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS 2 hours
Math 317: Hilbert Space
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.
SECTION A
A1. (a) Let^ V^ be a vector space over^ K(=^ R^ or^ C). (i) Define what is meant by a norm on V? [3] (ii) Show that if โ ยท โ is a norm on V then โฃโฃ โฃโuโ โ โvโ
โฃ โค โu โ vโ for all u, v โ V. [4] (b) Let X be a normed space. (i) What is meant by a closed subset A of X? [3] (ii) Show that if (xn) is a Cauchy sequence in X then (โxnโ) is a convergent sequence in R. [HINT: Use (a)(ii).] [4]
A2. Let Fb[0, 1] denote the vector space of all bounded complex valued functions defined on [0, 1] and let C[0, 1] denote the vecrtor space of continuous complex-valued functions on [0, 1]. (a) Show that โf โsup := sup{|f (t)| : t โ [0, 1]} defines a norm on Fb[0, 1]. [4] (b) Show that C[0, 1] is a subspace of Fb[0, 1]. [5] (c) Why is C[0, 1] a closed subspace of Fb[0, 1]? [5] [You may quote any relevant theorems in your justifications.] please turn over
SECTION A continued
A3. (a) Let p โ X and A โ X for a normed space X. Define what is meant by the distance dist(p, A), of p from A. [2] (b) Let X 1 be the normed space (R^2 , โ ยท โ 1 ) and let Xโ be the normed space (R^2 , โ ยท โโ), where โ(x, y)โ 1 := |x| + |y| and โ(x, y)โโ := max{|x|, |y|}. (i) Draw the unit spheres A 1 in X 1 , and Aโ in Xโ, [4] (ii) Let p = (1, 1) โ R^2. Find the following distances and sets of points (with respect to the corresponding norms): d 1 := dist(p, A 1 ); C 1 = {x โ A 1 : โx โ pโ 1 = d 1 }; dโ := dist(p, Aโ); Cโ := {x โ Aโ : โx โ pโโ = dโ}.
[6]
A4. (a) State and prove the Cauchy-Schwarz inequality for elements of an inner product space. [6] (b) Prove that (^) โซ ฯ 2 0
ex^ cos x dx โค
โฯ 2
eฯ^ โ 1
. [4]
please turn over
SECTION B continued
B3. (a) Let H be a complex Hilbert space. (i) Let x โ H. Show that the map
ฯ : H โ C, v โ ใx, vใ
is a bounded linear functional. [5] (ii) State the Riesz-Frยดechet Theorem. [5] (iii) What is meant by saying that a map T from H to H is a bounded operator? [5] (b) Show the following (i)
V f
(x) =
โซ (^) x 0 f (t) dt defines a bounded operator on the Hilbert space L^2 [0, 1]. [5] (ii) โV โ โค โ^12. [HINT: Use the Cauchy-Schwarz Inequality.] [5] (iii) โV โ โฅ โ^13. [HINT: Find โV f โ for f = 1.] [5]
end of exam
