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LANCASTER UNIVERSITY
2008 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^ x^ ∈^ X^ and^ E^ ⊂^ X^ for a normed space^ X. Define what is meant by the distance of x from E. [4] (b) Draw a sketch of the unit sphere S in X when X = `p R(2), for (i) p = 1, (ii) p = 2 and (iii) p = ∞. In case p = 2, find the distance of x = (3, 4) from S. [16]
A2. (a) What is the parallelogram law for the norm derived from an inner product?^ [4] (b) Show that the Banach space C[0, 1] (with sup-norm) is not a Hilbert space. [6] [Hint: Find elements f, g ∈ C[0, 1] such that the parallelogram law fails on these.]
A3. Let u, v ∈ H, for a Hilbert space H. (a) State the Cauchy–Schwartz inequality for u and v, and show that equality holds if u and v are linearly dependent. [7] (b) Suppose that H is a real Hilbert space. Given that equality holds in the Cauchy– Schwartz inequality if and only if the two vectors are linearly dependent, find precisely when the triangle inequality in H is an equality. [5] [Hint: ‖u + v‖^2 = (‖u‖ + ‖v‖)^2 implies 〈u, v〉 = ‖u‖‖v‖.]
A4. Let S ⊂ H for a Hilbert space H. (a) What is the orthogonal complement of S? [4] (b) Show that S⊥⊥^ ⊃ S. [4] please turn over 1
SECTION B
B1. (a) Define the terms (i) Banach space, (ii) bounded operator. [8] (b) (i) Explain carefully what ^1 and∞^ are and how their norms are defined. [5] (ii) Prove that the norm of ^1 does satisfy the norm axioms. [6] (iii) Sketch a proof that^1 is a Banach space. [6] (c) Let T : ^1 →^1 be the linear map defined by T z = (α 1 z 1 , α 2 z 2 ,.. .) for z = (z 1 , z 2 ,.. .) ∈ ^1 , where α ∈∞. Show that T is bounded and ‖T ‖ ≤ ‖α‖∞. [5]
B2. (a) Let U be a closed subspace of a Hilbert space H. Show that the following are equivalent, for vectors x ∈ H and x 0 ∈ U : (i) dist(x, U ) = ‖x 0 − x‖; (ii) (x − x 0 ) ∈ U ⊥. [15] (b) Let U be the subspace of L^2 [0, 2 π] linearly spanned by 1, cos t and sin t, and let f (t) = et. Find the function g in U nearest to f. [15] [Hint: You may use the identities ∫ (^2) π 0 es^ cos s ds =^12 (e^2 π^ − 1) and
∫ (^2) π 0 es^ sin s ds = − 2 1 (e^2 π^ − 1). ]
B3. (a) State and prove the Riesz–Fr´echet Theorem. [15] (b) Give an example of an inner product space X and a bounded linear functional ϕ on X such that there is no vector v ∈ X satisfying ϕ(x) = 〈v, x〉 for all x ∈ X. [15]
[Hint: Let X = c 00 , the space of complex sequences whose terms are eventually zero, viewed as a subspace of the Hilbert space ^2 , and let ϕ(x) = ∑∞ n=1 znxn where z = (z 1 , z 2 ,.. .) ∈^2 \ c 00. You may use the fact that c 00 is dense in `^2 .]
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