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FAN802S - FUNCTIONAL ANALYSIS - 2ND OPP - JANUARY 2025 |
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nAml BIA UnlVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoolof 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
QUALIFICATION CODE: 08BSHM
LEVEL: 8
COURSE:FUNCTIONALANALYSIS
COURSECODE: FAN802S
DATE: JANUARY 2025
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY: QUESTION PAPER
EXAMINER:
MODERATOR:
Dr SN NEOSS/-NGUETCHUE
Prof F. MASSAMBA
INSTRUCTIONS:
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in proofs and obtaining results.
3. All written work must be done in blue or black ink and sketches must be done in
pencil.
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 =/=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.
[3+2]
1-1-2. A Borel a-algebra on X.
[3]
1-1-3. A measurable space on X.
[1]
1-1-4. A measure on X.
[3]
1-1-5. A measure space on X.
[1]
1-2. Let EC JRa non-empty set. Show that F = {0,E, Ee, 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
~c.
Problem 2: [35 Marks]
Let (X, 11· 11)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'f'(n)}such that
[6]
1
Vk EN, llx'f'(k)- x'f'(k-1)11:S 2k-l
and show that
[6]
n
X'f'(n)= I)x'f'(k) - X'f'(k-1))+ X'f'(O),for any n 2: l.
k=l
2-2-3. Deduce from question 2-2-2 that the sequence {X<p(n)c}onverges.
[6]
2-2-4. Conclude that {xn} converges and therefore Xis 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,i),where X = C1[0, 1] and llflloo,1= sup lf(x)I + sup IJ'(x)I and also consider
xE[O,l]
xE[O,l]
(Y, II· lloo),where Y = C[0, 1].
3-1-1. Show that T = d~: X--+ Y is a bounded linear operator.
[7]
3-1-2. Show that T = ddx: D(T) £; Y--+ Y is an unbounded linear operator, where D(T) = C1[0, 1]. [10]
(Hint: use un(x) = sin(mrx)).
t, <c,o, } (t, r 3-2. We recall that /l,2 or /l,2 sometimes denoted /l,2 (N0 ) is the space of sequences defined by
l
e'- {X - (Xn)neo' lxnI'
No- Nu {O}' and llxll,,-
Ix.I'
Show that the following operators are linear and continuous and compute their norms.
3-2-1. T1: /l,2 --+/l,2 : T1 ((xn)n2'.0)= (Xn+1)n2'.0·
[9]
f 3-2-2. T2: L2 ([0, 1])--+ C: T2 (!) = 01x2 f(x)dx, where:
[12]
{!: oo} L2([0, 1]) =
f [0, 1]--+ JR:
1
0
lf(x)l2dx
<
(f and IIJIIL2=
1
0
lf(x)l2dx)
1
2
.
God bless you !!!