MA3G7 Functional Analysis I 2020 Practice Exam PDF

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MA3G7 Functional Analysis I 2020 Practice Exam...

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MA3G70

THE UNIVERSITY OF WARWICK THIRD YEAR EXAMINATION: APRIL 2020 FUNCTIONAL ANALYSIS I

Time Allowed: 3 hours Read all instructions carefully. Please note also the guidance you have received in advance on the departmental ‘Warwick Mathematics Exams 2020’ webpage. Calculators, wikipedia and interactive internet resources are not needed and are not permitted in this examination. You are not allowed to confer with other people. You may use module materials and resources from the module webpage. ANSWER COMPULSORY QUESTION 1 AND TWO FURTHER QUESTIONS out of the four optional questions 2, 3, 4 and 5. On completion of the assessment, you must upload your answer to the AEP as a single PDF document. You have an additional 45 minutes to make the upload, and instructions are available on the departmental ‘Warwick Mathematics Exams 2020’ webpage. You must not upload answers to more than 3 questions, including Question 1. If you do, you will only be given credit for your Question 1 and the first two other answers. The numbers in the margin indicate approximately how many marks are available for each part of a question. The compulsory question is worth twice the number of marks of each optional question. Note that the marks do not sum to 100.

COMPULSORY QUESTION 1.

a) Let V , W be vector spaces and A : V → W be a linear map. Show that dim(A(V )) ≤ dim V (as either a finite number or ∞). b) Let S be a set and (X, k· k) be a normed space. Let Y be the set which contains all functions f : S → X such that kf kY = sups∈S kf (s)k < ∞. Show that this equation defines a norm on Y . P ∞ c) Let ℓ1 (R) be the space of real sequences (xk )k=1 such that kxk1 = ∞ k=1 |xk | < ∞ and let kxk∞ = maxk∈N |xk |. (i) Show that the norms k · k1 and k · k∞ are not equivalent on ℓ1 (R). 1

(ii) Show that the space ℓ (R) with the norm k · k∞ is not complete. 1

Question 1 continued overleaf

[3]

[3]

[3] [3]

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d) Let X be a normed space and Y ⊂ X be a linear subspace. Show that if Y is open then Y = X. p e) Let H be a vector space over C with an inner product h·, ·i and let kxk = hx, xi be the natural norm on H. Show that |hx, yi| ≤ kxk kyk for all x, y ∈ H. P f) Let kxk1 = nk=1 |xn | for any x = (x1 , x2 , . . . , xn ) ∈ Rn . Show that the norm k · k1 cannot be obtained from any inner product on Rn .

[3] [3] [4]

g) Let H be a Hilbert space. Let P : H → H be a linear map such that P (P (x)) = P (x) for all x ∈ H and hP (x), yi = hx, P (y)i for all x, y ∈ H. Show that there is a closed linear subspace U ⊂ H such that P is an orthogonal projection onto U . [4] ∞ be an orthonormal sequence h) Let H be a complex Hilbert space and let (en )n=1 P∞ −α in H. Find for which real values of α the series n=1 n en converges in H . Justify your answer. 2

[3]

2

i) Let A : L [(0, 2π)] → L [(0, 2π)] be an operator defined by Z 2π (Af )(t) = cos(t) sin(s)f (s) ds . 0

(i) Show that the operator A is continuous.

[4]

(ii) Find all eigenvalues of A. Justify your answer.

[4]

(iii) Find the dimension of the range of A. Justify your answer.

[3]

OPTIONAL QUESTIONS 2. Let C[0, 1], the space of continuous functions, be equipped with the supremum norm. Let f ∈ C[0, 1]. a) Show that the sequence of polynomials n   X n k x (1 − x)n−k f (k/n). pn (x) = k k=0

converges to f in C[0, 1].

[10]

n   X n k x (1 − x)n−k = 1, You may use without proof the identities: k k=0 n   n   X n X n (k − nx)2 xk (1 − x)n−k = nx(1 − x). kxk (1 − x)n−k = nx, k k k=0 k=0

b) Deduce that the space C[0, 1] is separable. c) The k-th moment of f is defined to be Mk = for all integer k ≥ 0, then f = 0.

2

[5] R1 0

xk f (x) dx. Show that if Mk = 0 [5]

CONTINUED

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∞ be an orthonormal sequence in H . 3. Let H be a complex Hilbert space and let (en )n=1

a) State the definition of an orthonormal basis for H. P∞ ∞ is an |hx, en i|2 for every x ∈ H. Show that (en )n=1 b) Suppose that kxk2 = n=1 orthonormal basis for H.

[2] [4]

∞ be an orthonormal basis for H. Show that Parseval’s relation c) Let (en )n=1 ∞ X hx, yi = hx, en ihen , yi n=1

holds for all x, y ∈ H.

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d) Let {v1 , ..., vk } be a linearly independent subset of H and let S = span{v1 , ..., vk }. Show that there is an orthonormal basis {u1 , ..., uk } for S.

[5]

e) Suppose that H is separable. Show that there is an orthonormal basis for H.

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4. Let X be a Banach space and A : X → X be a bounded linear operator. a) Let {λ1 , ..., λn } be a set of distinct eigenvalues of A, and for each 1 ≤ k ≤ n let xk be an eigenvector corresponding to λk . Show that the set {x1 , ..., xn } is linearly independent.

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b) Let X be a Banach space and T : X → X be a bounded linear operator such that kT k < 1. Let S = I − T . Show that the operator S : X → X is bijective, S −1 is bounded and 1 kS −1 k ≤ . 1 − kT k [6] c) State the definition of R(A), the resolvent set of A.

[2]

d) Let λ be an eigenvalue of A. Show that λ ∈ / R(A).

[2]

e) Show that R(A) is open.

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3

CONTINUED

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∞ and (fn )∞ 5. Let H be an infinite-dimensional complex Hilbert space. Let (en )n=1 n=1 be ∞ orthonormal sequences in H. Let (αn )n=1 be a sequence of complex numbers, and define a linear operator T : H → H by

Tx =

∞ X

αn hx, en ifn

n=1

∞ is bounded; a) Show that T is bounded if and only if the sequence (αn )n=1

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b) Show that T is compact if and only if limn→∞ αn = 0;

[6]

c) Show that the adjoint operator T ∗ : H → H is given by T ∗x =

∞ X

αn hx, fn ien .

n=1

[3] d) Show that the image of T is dense in H if and only if Ker T ∗ = {0}. e) Suppose that αn =

1 n

[3]

for all n ∈ N. Show that the image of T is not closed.

Carefully explain your arguments. You may use any theorem from the lectures without proof.

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END

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