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FAN802S - FUNCTIONAL ANALYSIS - 1ST OPP - NOV 2022 |
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nAm I BIA un IVE RSITY
OF SCIEn CE Ano TECHn OLOGY
FACULTYOF HEALTH,APPLIEDSCIENCESAND NATURALRESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of Science in Applied Mathematics Honours
QUALIFICATION CODE: 08BSHM
COURSECODE: FAN802S
LEVEL: 8
COURSENAME: FUNCTIONALANALYSIS
SESSION:
DURATION:
NOVEMBER 2022
3H00
PAPER: THEORY
MARKS: 100
EXAMINER
MODERATOR:
FIRSTOPPORTUNITY-- QUESTION PAPER
Dr S.N. NEOSSINGUETCHUE
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.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPERCONSISTSOF 2 PAGES{Including this front page)
Attachments
None
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Problem 1: [35 Marks]
fl ' . . O if XE (Q,
1-1. Let f: JR
JRsuch that x 1---+ 1,
If X
L
<j_{Q.
Show that f is Borel-measurable.
[10]
(Hint: for any a E JR,consider E = {x E JR: f(x) < a} and show that J- 1 (E) E B(JR))
n00
1-2. Let (X, F) be a measurable space. Prove that if An E F, n E N, then An E F.
[5]
1-3.
Let D be a non-empty
set and Fa C P(rt), a E J an arbitrary
n=l
collection
of
O"-algebras on
n.
State
the definition of a O"-algebraand prove that
[4+6=10]
r·1 r._
J .-
I
r
J
a
aEI
is a a-algebra.
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,AUB
in terms of A\\ B, B \\ A, An B where necessary.)
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 JR.
[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 EN} U {n} is compact in E.
[8]
Problem 3: [35 Marks]
3-1. Use the convexity of x I-+ ex to prove the Arithmetic-Geometric Mean inequality:
[5]
\\:Ix, y > 0, and O < >-< 1, we have: x>-yi->. ::s>; -x+ (1 - >-)y.
3-2. Use the inequality in question 2-1. to prove Young's inequality:
[6]
a/3::s;-apP
+ -/qJ, q \\:/a,/3>
0,
,
wnere p,q
E (1,oo):
l
p-
+
-q1
=
1.
3-3. Use the result in question 3-2. to prove Holder's inequality:
[7]
n
(n
) 1/p ( n
\\ 1/q
~IXiYil::::; ~lxilP
~IYilq) ,\\:Ix= (xi),y= (yi) EJRn, p,q as above.
3-4. Consider (X, II·ll00 ,1), where X = C1 [0, 1] and llfll00 ,1 = sup IJ(x)I+ sup lf'(x)I and also consider
xE(O,l]
xE(O,l]
(Y, II·!loo)w, here Y = C[O,1].
T d: 3-4-1. Show that = X Y is a bounded linear operator.
[7]
3-4-2. 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)).
God bless you !!!