MAT 324 Real Analysis
Fall 2014
Final Exam (Solutions) December 12, 2014
NAME:
Please turn o your cell phone and put it away. You are NOT allowed to use a
calculator or an electronic device.
Please show your work! To receive full credit, you must explain
MAT324: Real Analysis Fall 2014
Assignment 8 Solutions
Problem 1: Compute
y sin(x)exy dxdy,
(0,)(0,1)
and explain why Fubinis theorem is applicable.
Solution. Notice that
|y sin(x)exy |dxdy
yexy dxdy
(0,)(0,1)
(0,)(0,1)
The integrand on the right-hand si
MAT 324 Real Analysis
Fall 2014
Midterm October 23, 2014
NAME:
Please turn o your cell phone and put it away. You are NOT allowed to use a
calculator.
Please show your work! To receive full credit, you must explain your reasoning and
neatly write the step
MAT324: Real Analysis Fall 2014
Assignment 8
Due Thursday, December 4, in class.
Problem 1: Compute
y sin(x)exy dxdy,
(0,)(0,1)
and explain why Fubinis theorem is applicable.
Problem 2: Let 1 , 2 and be measures on a measurable space (X, F). Show that if
MAT324: Real Analysis Fall 2014
Assignment 7
Due Thursday, November 20, in class.
Problem 1: Suppose f L2 (R) L4 (R). Prove that f also belongs to L3 (R).
Problem 2: Determine if the following functions belong to L (R).
a) f (x) =
1
x2 (0,n]
b) f (x) =
1
MAT324: Real Analysis Fall 2014
Assignment 7 Solutions
Problem 1: Suppose f L2 (R) L4 (R). Prove that f also belongs to L3 (R).
Solution. Notice that f L2 (R) L4 (R) implies |f | L2 (R), and f 2 L2 (R). By Hlders
o
inequality,
|f |f 2 dx
R |f |
R
2 dx
1
2
MAT324: Real Analysis Fall 2014
Assignment 6 Solutions
Problem 1: Which of the following statements are true and which are false? Explain.
a) L1 (R) L2 (R)
b) L2 (R) L1 (R)
c) L1 [3, 5] L2 [3, 5]
d) L2 [3, 5] L1 [3, 5]
Solution.
a) The statement is false.
MAT324: Real Analysis Fall 2014
Assignment 6
Due Thursday, November 6, in class.
Problem 1: Which of the following statements are true and which are false? Explain.
a) L1 (R) L2 (R)
b) L2 (R) L1 (R)
c) L1 [3, 5] L2 [3, 5]
d) L2 [3, 5] L1 [3, 5]
Problem 2:
MAT324: Real Analysis Fall 2014
Assignment 5 Solutions
Problem 1: Compute the following limits if they exist and justify the calculations:
a) lim
1+
n 0
x
n
n
sin
2 2
n2 xen x
dx.
1+x
n2 xen x
dx.
1+ nx
b) lim
n 0
c) lim
n 1
x
dx
n
2 2
Solution.
a) The i
MAT324: Real Analysis Fall 2014
Assignment 5
Due Tuesday, October 21, in class.
Problem 1: Compute the following limits if they exist and justify the calculations:
a) lim
1+
n 0
n
x
n
2 2
n2 xen x
dx.
1+x
n2 xen x
dx.
1+ nx
b) lim
n 0
c) lim
n 1
x
dx
n
s
MAT324: Real Analysis Fall 2014
Assignment 4 Solutions
Problem 1: Let C [0, 1] be the Cantor middle-thirds set. Suppose that f : [0, 1] R is dened
by f (x) = 0 for x C and f (x) = k for all x in each interval of length 3k which has been removed
from [0, 1
MAT324: Real Analysis Fall 2014
Assignment 4
Due Thursday, October 9, in class.
Problem 1: Let C [0, 1] be the Cantor middle-thirds set. Suppose that f : [0, 1] R is dened
by f (x) = 0 for x C and f (x) = k for all x in each interval of length 3k which ha
MAT324: Real Analysis Fall 2014
Assignment 1 Solutions
Problem 1: Let C be the Cantor middle-thirds set constructed in the textbook. Show that C is
compact, uncountable, and a null set.
Solution. The textbook proves that C is a null set (page 19). To chec
MAT324: Real Analysis Fall 2014
Assignment 2 Solutions
Problem 1: Construct a Cantor-like closed set C [0, 1] so that at the k th stage of the construction
one removes 2k1 centrally situated open intervals each of length k , with
1
+2
2
+ . . . + 2k1
Supp
MAT324: Real Analysis Fall 2014
Assignment 1
Due Tuesday, September 16, in class.
Problem 1: Let C be the Cantor middle-thirds set constructed in the textbook. Show that C is
compact, uncountable, and a null set.
Problem 2: Let A be the subset of [0, 1] w
MAT324: Real Analysis Fall 2014
Assignment 3
Due Thursday, October 2, in class.
Problem 1: Suppose that, for each rational number q, the set cfw_x | f (x) > q is measurable. Can
we conclude that f is measurable?
Problem 2:
dened by
Suppose f, g : E R are
MAT324: Real Analysis Fall 2014
Assignment 2
Due Thursday, September 25, in class.
Problem 1: Construct a Cantor-like closed set C [0, 1] so that at the k th stage of the construction
one removes 2k1 centrally situated open intervals each of length k , wi
MAT324: Real Analysis Fall 2014
Assignment 3 Solutions
Problem 1: Suppose that, for each rational number q, the set cfw_x | f (x) > q is measurable. Can
we conclude that f is measurable?
Solution. Yes, it is true. Use the density of Q in R to show that fo
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