The Man Who Saved Capitalism
Abbas P. Grammy
John Maynard Keynes (pronounced to rhyme with rains) was the most influential
economist of the twentieth century. His pioneering work, The General Theory of
Employment, Interest, and Money (1936) has led to the
Supplements to the Exercises in Chapters 1-7 of Walter Rudins
Principles of Mathematical Analysis, Third Edition
by George M. Bergman
This packet contains both additional exercises relating to the material in Chapters 1-7 of Rudin, and
information on Rudi
ANALYSIS HW 11
ALEX GITTENS
Problems
Exercise 16, pg. 291. If k 2 and = [p0 , p1 , . . .] is an oriented ane k -simplex, prove that 2 = 0,
directly from the denition of the boundary operator . Deduce from this that 2 = 0 for every chain .
Proof. By deniti
ANALYSIS HW 6
ALEX GITTENS
Problems
Exercise 3, pg. 239. Assume A L(X, Y ) and Ax = 0 only when x = 0. Prove that A is then 1-1.
Proof. Let x, y X be such that A(x) = A(y ), then A(x y ) = A(x) A(y ) = 0, so x y = 0 x = y .
Since A(x) = A(y ) x = y , A is
Math 118C Homework 5 Solutions
Charles Martin
May 12, 2009
1. Give an example of A, B R with A = B and d(A, B ) = 0.
Any sets A B where B \ A has measure zero will suce. For example, take A = R \ Q and B = R.
2. Prove that if f : R R is continuous then f
Math 118C Homework 6 Solutions
Charles Martin
May 20, 2009
Throughout the following, we use A to denote the characteristic function of A.
1. If f 0 and
E
f d = 0, prove that f = 0 almost everywhere on E .
For n N dene An = cfw_x : f (x) > 1/n, which is a
Math 118C Homework 4 Solutions
Charles Martin
April 28, 2009
1. If is an additive set function, prove that () = 0.
Denote the underlying ring by G. Notice that, per our tacit assumptions on set functions, (A) must be
nite for some A G.
Let A G be such tha
Math 118C Homework 7 Solutions
Charles Martin
May 20, 2009
1. If f L1 () and g is bounded and measurable on E , prove that f g L1 ().
Let M > 0 be such that |g | M . Then
|f g | d
X
M |f | d < .
X
2. Let g (x) = (1/2,1] . Dene f2k (x) = g (x) and f2k+1 (
Math 118C Homework 3 Solutions
Charles Martin
April 21, 2009
9.19 Show that the system of equations
3x + y z + u2 = 0
x y + 2z + u = 0
2x + 2y 3z + 2u = 0
can be solved in terms of x, in terms of y , in terms of z , but not in terms of u.
For k = 1, 2, 3
Math 118C Homework 2 Solutions
Charles Martin
April 16, 2009
9.11 If f and g are dierentiable functions on Rn , prove that
wherever f = 0.
(f g ) = f g + g f and that
(1/f ) = f 2 f
For i = 1, 2, . . . , n we have
Di (f g )ei = [f Di g + gDi f ] ei = f (D
ANALYSIS HW 8
ALEX GITTENS
Problems
Exercise 16, pg. 240. Show that the continuity of f at the point a is needed in the inverse function
theorem, even in the case n = 1: If
1
f (t) = t + 2t2 sin
t
for t = 0, and f (0) = 0, then f (0) = 1, f is bounded on
ANALYSIS HW 9
ALEX GITTENS
Problems
Exercise 9, pg. 290. Dene (x, y ) = T (r, ) on the rectangle
0 r a,
0 2
x = r cos ,
y = r sin .
by the equations
Show that T maps this rectangle onto the closed disc D with center at (0, 0) and radius a, that T is one-t
ANALYSIS HW 6
ALEX GITTENS
Theorems from the Book
Theorem 1 (9.17). Suppose f maps an open set E Rn into Rm , and f is dierentiable at a point x E .
Then the partial derivatives (Dj fi )(x) exist, and
m
f (x)ej =
(1 j n),
(Dj fi )(x)ui
i=1
where cfw_e1 ,
ANALYSIS HW 9
ALEX GITTENS
Problems
Exercise 5, pg. 289. Formulate and prove an analogue of Theorem 10.8, in which K is a compact subset
of an arbitrary metric space. (Replace the function i that occur in the proof of Theorem 10.8 by functions
of the type
Chapter 6
Convexity and differentiability.
6.1 Convex functions of one variable.
Proposition 6.1 Let f be a real-valued function dened on (a, b) R. The following
are equivalent:
1. f is convex on (a, b)
2.
3.
f (t1 ) f (t0 )
f (t2 ) f (t0 )
for all t0 , t
MATH 618 (SPRING 2010, PHILLIPS): SOLUTIONS TO
HOMEWORK 3
1. Rudin Chapter 9 Problem 2
I have restated the problem in labelled parts for convenience.
(a) Compute the Fourier transform of the characteristic function of an interval.
(b) For n Z>0 let gn be
Self-Assessment Questions Math 118A, Fall
2009
1. Every rational number can be written in the form x = m/n, where
n > 0, and m and n are integers without any comon divisors. When
x = 0, we take n = 1. Consider the functions f dened on R by
f (x ) =
0 if x
Homework 2 Math 118B, Winter 2010
Due on Thursday, January 28, 2010
1. Suppose increases on [a, b], a x0 b, is continuous at x0 , f (x0 ) =
1, and f (x) = 0 if x = x0 . Prove that f R() and that f d = 0.
b
a
2. Suppose that f 0, f is continuous on [a, b],
Homework 5 Math 118B, Winter 2010
Due on Thursday, March 4th, 2010
1. Show that if f 0 and if f is monotonically decreasing, and if
n
n
cn =
k=1
f (k )
f (x) dx,
1
then
lim cn
n
exists.
2. Let n (x) be positive-valued and continuous for all x [1, 1], wit
Self-Assessment Questions Math 118A, Fall
2009
1. Evaluate the limit
n
lim n
n
e
1
1+
n
n
.
without using LHospitals rule.
2. Prove that
n
1
1+
n
lim
n
exists using the following steps:
(a) Prove that
n
an =
1
1+
n
1
1
n
n
bn =
and
are both increasing se
Homework 1 Math 118B, Winter 2010
Due on Thursday, January 14, 2010
1. Let f be dened for all real x, and suppose that M > 0 and > 0
such that
|f (x) f (y )| M |x y |1+ , x, y R.
Prove that f is constant.
2. Suppose that f (x) > 0 in (a, b). Prove that f
Homework 3 Math 118B, Winter 2010
Due on Thursday, February 4, 2010
1. Let be a xed increasing function on [a, b]. For u R(), dene
1/2
b
u
2
2
=
|u| d
.
a
Suppose f, g, h R(), and prove the triangle inequality
f h
2
f g
2
+ g h 2,
as a consequence of the
Self-Assessment Questions Math 118A, Fall
2009
1. For any sequence cfw_cn of positive numbers,
lim inf
n
cn+1
cn+1
lim inf n cn lim sup n cn lim sup
.
n
cn
cn
n
n
2. For any two real sequences cfw_an , cfw_bn , prove that
lim sup(an + bn ) lim sup an +
Homework 6 Math 118B, Winter 2010
Due on Thursday, March 11th, 2010
1. Let
0
sin2
fn (x) =
0
x
1
n+1
1
x < n+1 ,
1
x n,
1
x > n.
Show that cfw_fn converges to a continuous function, but not uniformly.
Use the series
fn to show that absolute convergence,
Homework 1 Math 118B, Winter 2010
Due on Thursday, January 14, 2010
1. Let f be dened for all real x, and suppose that M > 0 and > 0
such that
|f (x) f (y )| M |x y |1+ , x, y R.
Prove that f is constant.
Proof: If we divide by |x y |, we get
f (x) f (y )