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Qualifying II

The document contains instructions for an exam on real and functional analysis consisting of 8 problems. Students must complete 5 problems, with each problem worth 20 points. The problems cover topics such as convergence of sequences and series, properties of continuous and measurable functions, integration, and convergence in function spaces. Good luck is wished to the students taking the exam.

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72 views2 pages

Qualifying II

The document contains instructions for an exam on real and functional analysis consisting of 8 problems. Students must complete 5 problems, with each problem worth 20 points. The problems cover topics such as convergence of sequences and series, properties of continuous and measurable functions, integration, and convergence in function spaces. Good luck is wished to the students taking the exam.

Uploaded by

getachewbonga09
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Adama Science and Technology University

School of Applied Natural Science


Department of Applied Mathematics
PhD Comprehensive Qualifying Exam for
Real and Functional Analysis
Date: 01, 11, 2019
Time : 3:00 Hrs

Code:

INSTRUCTIONS:

• There are eight problems and do any five of them.

• Each problem worth 20 points.

• If you answer more than five problems, only the first five will be marked.

• Use a separate sheet of paper for each problem, and write your Code on each answer
sheet.

• Give clear reasoning.

• State clearly which theorems you are using.

• You should not cite anything else such as examples, exercises, or problems.

• Cross out the parts you do not want to be marked.

• µ is the Lebesgue measure.

∼ Good Luck ∼
1
1. Let {an } be a sequence of real numbers such that |an − an+1 | ≤ n(n+1)
, for each n ∈ N.
Show that {an } converges.

2. Prove that the series ∞ √


X n + cos nx
n=1
x2 + n 2
converges uniformly over R.

3. Let f : [0, 1] → R be a continuous function.


Z 1
(a) Show that lim xn f (x)dx = 0.
n→∞ 0
R1
(b) If 0
f 2 (x)dx = 0, then show that f (x) = 0 for all x ∈ [0, 1].

4. If E is a nonempty subset of a metric space X, define the distance from x to E by


d(x, E) = inf d(x, y). Prove that
y∈E

(a) If d(x, E) = 0 then x ∈ Ē.


(b) d(x, E) is a uniformly continuous function.

5. Let E ⊂ [0, 1] such that µ(E) > 0. Show that for each 0 <  < 1, there is an open
interval I satisfying
µ(E ∩ I)
< .
µ(I)
6. If f ≥ 0, fn are measurable functions, then prove that
∞
Z X ∞ Z
X
fn dµ = f dµ.
E n=1 n=1 E

√
7. Suppose µ(X) = 2, f and g are non-negative measurable functions such that f g ≥ 1
a.e.
Prove that Z Z
f dµ gdµ ≥ 2.
X X

8. Define gn (x) = nχ[0,n−3 ] (x).


R1
(a) Show that if f ∈ L2 ([0, 1]), then f (x)gn (x)dx → 0 as n → ∞.
0
R1
(b) Find a function f ∈ L1 ([0, 1]) such that 0 f (x)gn (x)dx does not converge to 0 as
n → ∞.

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