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FAN802S - FUNCTIONAL ANALYSIS - 2ND OPP - JAN 2023 |
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n Am I BI A u n IVER s I TY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,APPLIEDSCIENCESAND NATURALRESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of Science in Applied Mathematics Honours
QUALIFICATION CODE: 08BSHM
COURSE CODE: FAN802S
LEVEL: 8
COURSE NAME:
FUNCTIONAL ANALYSIS
SESSION:
DURATION:
JANUARY 2023
3H00
PAPER: THEORY
MARKS: 100
SECONDOPPORTUNITY/SUPPLEMENTARY-- QUESTION PAPER
EXAMINER
Dr S.N. NEOSSNI GUETCHUE
MODERATOR:
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 PAPER CONSISTS OF 2 PAGES (Including this front page)
Attachments
None
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2 Page 2 |
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Problem 1: [45 Marks]
0 if XE (Q,
1-1.
Let f:
JR-+ JRsuch that
x
1--r {
'
1,
.
1fxr: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 1-1(E) E B(JR))
II00
1-2. Let (X, F) be a measurable space. Prove that if An E F, n EN, then An E F.
[5]
n=l
1-3. Let n be a non-empty set and Fa c P(n), a E J an arbitrary collection of o--algebras on n. State
.:·:-r,.:: the definition of a o--algebra and prove that
I J .-
Ja
is a O"-algebra.
[4+6=10]
aEI
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, B EA, 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) = co; (ii) µ(A),µ(B) < oo and then express A,B,AUB
in terms of A \\ B, B \\ A, A n B where necessary.)
1-5. Show that the following Dirichlet function is Lebesgue integrable but not Riemann integrable [10]
X := li1Qn[o,1[]:O1, ]-+ JR
1, if XE (Q
1 X 1---7{ 0, if X (Q
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 1--r ex to prove the Arithmetic-Geometric Mean inequality:
[5]
'-ix,y > 0, and O < .A< 1, we have: x>-y1->- ,\\x + (l - ,\\)y.
3-2. Use the inequality in question 2-1. to prove Young's inequality:
[6]
a/3 '5:-apP + -fqP,
'-ia,/3 >
0,
.
where
p,q
E (1,oo):
p-l +-
1
q
=
1.
3-3. Use the result in question 3-2. to prove I-folder's inequality:
[7]
8 8 8 n
(n
) 1/p ( n
, 1/q
lx,y,I:o'. Ix,IP , IYI,') ,\\Ix = (x,), y = (y,) E lilt", p, q asabove .
3-4. Consider (X, II· lloo,1),where X = C1[0,1] and 111001,11 = sup lf(x)I + sup lf'(x)I and also consider
xE[O,l]
xE[O,l]
(Y, II· lloo),where Y = C[0, 1].
3-4-1. Show that T = -ddx : X -+ Y is a bounded linear operator.
[7]
3-4-2. Show that T = d~: D(T) s;;Y-+ Y is an unbounded linear operator, where D(T) = C1 [0,1]. [10]
(Hint: use un(x) = sin(mrx)).
God bless you !!!