Math 479: HW 9
Due Thursday, April 2nd, 2015
Problem 1
Let X, Y be two vector spaces, and A L(X, Y ). Let null(A) cfw_x X : Ax = 0. Show A is 1-1 if and
only if null(A) = cfw_0.
(Note: null(A) is sometimes also called kernel of A. So this is saying that a
NAME:
Binghamton University
REAL ANALYSIS: MATH 479
SPRING 2015
EXAM 2
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- 1 a). [4 pts] Finish the denition: a family .7 of real valued functions f : [(1, b] + R is
equicontinuous if and only if V E > O 3 g > 0 Star.
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Math 479: HW 10
Due Thursday, April 16th, 2015
Problem 1
Why can we assume f : Rn R in the proof of Theorem 9.21?
Problem 2
Suppose f : Rn Rn is a local dieomorphism. Show f (x) : Rn Rn is an isomorphism for all x Rn ,
i.e., show f (x) is invertible for a
Introduction to measure theory and construction of the
Lebesque measure
Intro to measure theory: based on baby Rudin
Construction of the Lebesgue measure: based on Jones (Lebesgue
Integration on Euclidean Space)
Real Analysis 2, Spring 2015
Binghamton Uni
Math 479: HW 7
Due Thursday, 3/19/2015
Problem 7.1
Prove Theorem 7.17 under the extra assumption that fn are continuous.
Problem 7.2
a) Finish the proof of the lemma from class: compact metric space K is separable. Show the set S we
constructed is dense.
Math 479: HW 5
Due Thursday, 2/26/2015
Problem 5.1
5.1a) Let f be continuous on [a, b]. Show there exists c [a, b] such that
f (c) =
1
ba
b
f (x)dx.
a
5.2b) Let f, g be continuous on [a, b], and g be a function that does not change a sign on [a, b] (i.e.,
Math 479: HW 1
Due Thursday, January 29th, 2015
Problem 1.1
Let f : R R be continuous. Show
a) The set cfw_x R : f (x) = 0 is closed.
b) The set cfw_x R : f (x) > 0 is open.
c) Show if f (x0 ) > 0 for some x0 R, then there exists r > 0 such that f (x) > 0
Math 479: HW 2
Due Thursday, 2/5/2015
Problem 2.1
Answer the following questions for each of the functions below.
1. Is f continuous on (, 0) (0, )? Quote a theorem to justify your answer.
2. Use the denition of continuity to prove or disprove: f is conti
Math 479: HW 3
Due Thursday, 2/12/2015
Problem 3.1
p. 114, # 7. Make sure to justify why
f (t)
g(t)
is well dened, when considering limtx
f (t)
g(t) .
Problem 3.2
p. 117, # 22 a, b, c
Problem 3.3
Let f : [0, 2] R2 be given by f (t) = (2 cos t, 2 sin t).
1
Math 479: HW 8
Due Thursday, 3/26/2015
Problem 8.1
Let C and be two xed positive real numbers. Dene K to be the set of all real-valued functions on [0, 1]
satisfying
|f (x) f (y)| C|x y| , x, y [0, 1].
Is K equicontinuous? Closed? Bounded? Compact?
Proble
Math 479: HW 6
Due Thursday, 3/5/2015
Problem 6.1
p. 165 # 1
Problem 6.2
p. 165 # 2
Problem 6.3
Recall a normed vector space is a vector space X together with a mapping | | satisfying
0) | | : X [0, )
1) |x| = |x| for every x X, and R.
2) |x| = 0 if and o
Math 479: HW 13
Due Thursday, May 7th, 2015
Problem 13.1
p. 332 # 1
Problem 13.2
Show properties a-d stated in Remark 11.23 on p. 315
Problem 13.3
Recall the Cantor set C (p. 41). Cantor set is uncountable. Compute m(C).
Problem 13.4
Consider the followin
Math 479: HW 12
Due Wednesday, April 29th, 2015
Problem 1
Show a closed line segment has n dimensional Lebesgue measure equal to 0 for n 2.
Show Q has Lebesgue measure 0.
Problem 2
Let G be an open subset of Rn . Show m (G) = m (G) = m(G) where m(G) is
Math 479: HW 11
Due Thursday, April 23rd, 2015
Problem 1
Consider the following system of equations:
x2 + x2 + y 2 = 1,
1
2
x1 + x2 + y = 1.
Can we apply the Implicit Function Theorem to solve for x = (x1 , x2 ) in terms of y in the neighborhood
of (1, 0,
Math 479: HW 4
Due Thursday, 2/19/2015
Problem 4.1
Let x0 [a, b] and f (x0 ) = 1, and f (x) = 0 for x = x0 . Show f R and that
f dx = 0.
Problem 4.2
p. 138 2
Problem 4.3
a) Prove Theorem 6.7 (with (x) = x).
b) If f is Riemann integrable on [a, b], show cf
Binghamton University
REAL ANALYSIS: MATH 479
SPRING 2015
EXAM 1
NAME: 35)] Lia {9 M f;
n Points 1 a). [4 pts] State the Intermediate Value Theorem.
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