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FAN802S - FUNCTIONAL ANALYSIS - 1ST OPP - NOVEMBER 2024 |
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nAml BIA UnlVERSITY
OF SCIEnCEAnDTECHnOLOGY
FacultoyfHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
PrivateBag 13388
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:FUNCTION ANALYSIS
COURSECODE: FAN802S
DATE: NOVEMBER 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 94
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY: QUESTION PAPER
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: [34 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.
1-1-2. A Borel a-algebra on X.
1-1-3. A measurable space on X.
1-1-4. A measure on X.
1-1-5. A measure space on X.
1-2. Let Ea non-empty set and A E P(E). Determine the a-algebra generated by C = {A}.
1-3. Let£ be a a-algebra on X, and X 0 C X.
1-3-1. Show that £0 = {An X 0 IAE £} is a a-algebra on X 0 .
1-3-2. Show that a(£) = £.
1-4. Let K, K,' c P(X). Show that, if K, c K,' c a(K), then a(K') = a(K).
Problem 2: [31 Marks]
t, 00}, (t, r We recall that /l,2 or /l,2 sometimes denoted /l,2(N0) is the space of sequences defined by
f' = { X = (x.)neo: Ix.I'<
No= N LJ{O}, and llxll,=,
I
Ix.I'
[2+2]
[2]
[l]
[2]
[l]
[6]
[7]
[5]
[6]
2-1. We assume that H = /l,2 is a complete space for the norm associated with (·, ·)H · Show that H = /l,2
is a Hilbert space with respect to
[10]
L00
(x, Y)H = XnYn
n=l
2-2. Show that the following operators are linear and continuous and compute their norms.
2-2-1. T1: R2 -t R2 : T1 ((xn)n~o) = (xn+1)n~O·
[9]
f 2-2-2.
T2:
L2 ([0, l]) -t
<C:T2 (J) =
1
0
x
2
f(x)dx,
where:
[12]
{!: L2 ([0, l]) =
l
[0, l] -t
f IR:
1
0
lf(x)l2dx
< oo} and IIJIIL=2(f01 lf(x)l2dx)
2
.
Problem 3: [29 :tvlarks]
3-1. State the Monotone Convergence Theorem (MCT) and the Dominated Convergent Theorem (DCT)
respectively.
[6]
3-2. Show that the function f: (0, oo) -t IR,f(x) := esXin- x1 ,\\:/x > 0, is Lebesgue integrable on [0,oo]. [6]
00
3-3. Show that for any x > 0, we have f(x) = I::e-nxsinx.
[5]
n=l
l + 3-4.
Deduce that
·oo sinx
--
=
. 0 ex - 1 n=l
-
n
2--
1
•
1
[12]
God bless you !!!