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FAN802S - FUNCTIONAL ANALYSIS - 1ST OPP - NOVEMBER 2023 |
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nAmlBIA UntVERSITY
OF SCIEnCEAnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: •264 612072913
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: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: EXAMINATION QUESTION PAPER
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: [27 Marks]
1-1. Let X -f 0. Give the definition of the following concepts:
1-1-1. A a-algebra on X and a a-algebra generated by a family C of subsets of X.
1-1-2. A Borel a-algebra on X.
[3+2]
[3]
1-1-3. A measurable space on X.
[l]
1-1-4. A measure on X.
[3]
1-1-5. A measure space on X.
[l]
1-2. Let EC JRa non-empty set. Show that F = {0, E, EC,JR}is the a-algebra of subsets of JRgenerated
by {E}.
[9]
1-3. Let X = {l, 2, 3,4} and consider C = {{1},{2,3}} C P(X). Determine a(C) the a-algebra generated
by C.
[5]
Problem 2: [35 Marks]
Let (X, II· II) be a normed space.
2-1. Assume that X is a Banach space.
Show that any absolutely summable series is summable.
[6]
2-2. Now we assume that X is a normed space in which any absolutely summable series is summable.
2-2-1. Let {xn} be a Cauchy sequence in X. Show that if {xn} has a convergent subsequence {xnk}, {xn}
converges to the same limit.
[6]
2-2-2. Show that we can construct a subsequence {X,p(n)} such that
[6]
1
Vk E N, llx,p(k)- X,p(k-1) II 2k-l
and show that
[6]
n
X,p(n) = I)x<p(k) - X,p(k-1))+ X,p(O), for any n 2: l.
k=l
2-2-3. Deduce from question 2-2-2 that the sequence {X,p(n)} converges.
[6]
2-2-4. Conclude that {Xn} converges and therefore X is a Banach space.
[3]
2-3. What is the general rule that you can establish from the main results obtained above.
[2]
Problem 3: [38 Marks]
3-1. Consider (X, II· lloo,1),where X = C1[0,l] and 111001,11= sup IJ(x)I + sup IJ'(x)I and also consider
xE[0,l]
xE[0,l]
! : (Y, II· lloo),where Y = C[O,l].
3-1-1. Show that T = X Y is a bounded linear operator.
[7]
3-1-2. Show that T = d~: D(T) £; Y
(Hint: use Un(x) = sin(mrx)).
Y is an unbounded linear operator, where D(T) = C1[0,l]. [10]
3-2. We recall that £2 or £2 sometimes denoted £2 (N0 ) is the space of sequences defined by
t, oo•} (t, t' {x (xn)neo, lx.12 <
No~ NU (0), and llxlll'
I
Ix.I')'
Show that the following operators are linear and continuous and compute their norms.
3-2-1. T1: £2 f2 : T1 ((xn)n;::o)= (xn+i)ne::O·
[9]
f 3-2-2. T2: £ 2([0,l]) C: T2(!) = 01x2J(x)dx, where:
[12]
{!: oo} L2([0,l]) =
J [0,l]
JR:
1
0
IJ(x)J2 dx
<
l
and
11!11=£2(J01
IJ(x)l2dx)
2
.
God bless you !!!