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FAN802S - FUNCTIONAL ANALYSIS - 2ND OPP - JANUARY 2024 |
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.-
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nAm I BI A UnlVERSITY
-~
OF SCIEnCEAno TECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: •264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCE IN APPLIED MATHEMATICS HONOURS
QUALIFICATIONCODE: 08BSHM
LEVEL:8
COURSE:FUNCTIONAL ANALYSIS
COURSECODE: FAN802S
DATE: JANUARY 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/ SUPPLEMENTARY: EXAMINATION QUESTION PAPER
EXAMINER:
MODERATOR:
Dr S.N. NEOSSI-NGUETCHUE
Prof F. MASSAMBA
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Show clearly all the steps used in the calculations.
6. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
ATTACHEMENTS
None
This paper consists of 2 pages including this front page
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Problem 1: [45 Marks]
1-1. Let f: IR
IR such that
xM
0,
{ 1,
if XE (Ql,
if X (Ql.
Show that f is Borel-measurable.
[10]
(Hint: for any a E IR, consider E = {x E IR: j(x) < a} and show that 1-1(E) E B(IR))
n00
1-2. Let (X, F) be a measurable space. Prove that if An E F, n EN, then An E F.
[5]
1-3. Let n be a non-empty set and Fo. c P(n), a E I an arbitrary
the definition of a o--algebra and prove that
n F := Fo. is a o--algebra.
n=l
collection of o--algebras on n. State
[4+6=10]
o.El
1-4. Let (X, A,µ) be a measure space.
(i) What does it mean that (X, A,µ) be a measure space?
[3]
(ii) Show that for any A, BE A, we have the equality: µ(AU B) +µ(An B) =µ(A)+ µ(B).
[7]
(Hint: Consider two cases: (i) µ(A)= oo or µ(B) = oo; (ii) µ(A), µ(B) < oo and then express A, B, AU B
in terms of A\\ B, B \\ A, An B where necessary.)
1-5. Show that the following Dirichlet function is Lebesgue integrable but not Riemann integrable [10]
X := ].Qn(o,1J : [0,l] ->-IR
X f-r
l,
{ 0,
ifxE(Ql
if X (Ql
Problem 2: [20 Marks]
2-1. Define what is a compact set in a topological space.
[3]
2-2. Show that (0, 1] is not a compact set for usual topology of IR.
[9]
2-3. Let E be a Hausdorff topological space and {an}nEN a sequence of elements of E converging to a.
Show that K = {anln E N} U {n} is compact in E.
[8]
Problem 3: [35 Marks]
3-1. Use the convexity of x H ex to prove the Arithmetic-Geometric Mean inequality:
[5]
Vx, y > 0, and 0 < >-< 1, we have: x>-yl->. :'.S>: -x+ (1 - >-)y.
3-2. Use the inequality in question 2-1 to prove Young's inequality:
[6]
aP $q
a$ :'.S:-p +-,qVa,$>
0, where p,q E (l,oo):
1
-p
+
-q1
=
1.
3-3. Use the result in question 3-2 to prove Holder's inequality:
t t t [x;y;[,S (
[x;[') J/p (
119
[y,[9) , I/ x = (x,), y = (y;)E IRn,p, q as above .
[7]
3-4. Consider (X, II · 1010 ,1), where X = C1[0, l] and llflloo,=1 sup lf(x)I + sup lf'(x)I and also consider
xE[O,l)
xE[O,l)
= (Y, II · !100 ), where Y C[0, l].
3-4-1. Show that T = ,d.!x!:_X: Y is a bounded linear operator.
[7]
3-4-2. Show that T = .d.xE_D: (T)
(Hint: use un(x) = sin(mrx)).
Y is an unbounded linear operator, where D(T) = C1 [0, 1]. [10]
God bless you ! ! !