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This is the Past Exam of Hilbert Space which includes Unit Sphere, Normed Space, Distance, Norm Derived, Hilbert Space, Parallelogram, Cauchy Schwartz Inequality, Equality Holds, Linearly Dependent etc. Key important points are: Unit Sphere, Normed Space, Distance, Norm Derived, Hilbert Space, Parallelogram, Cauchy Schwartz Inequality, Equality Holds, Linearly Dependent, Triangle Inequality
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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
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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