2
1
SERIES
Analysis Study Guide
Sequences
Def: An ordered field is a field F and total order < (for all x, y, z ∈ F ):
(i) x < y, y < x or x = y,
(ii) x < y, y < z ⇒ x < z
(iii) x < y ⇒ x + z < y + z
(iv) 0 < y, x ⇒ 0 < xy.1
Def: The Archimedean property on an ordered field F is ∀x, y ∈ F, x, y > 0, there exists N ∈ N such that n · x > y.2
a
3
Fact: ab = dc ⇒ ra+sc
rb+sd = b for all a, b, c, d, r, s ∈ Z with rb + sd 6= 0.
Def: A real number L is the limit of a sequence of real numbers (an )∞
n=1 if for every ε > 0, there is an N ∈ N such that
|an − L| < ε for all n ≥ N . Then, (an ) converges to L.4
Thm: “Squeeze Theorem” Suppose three sequences (an ), (bn ), (cn ) satisfy an ≤ bn ≤ cn and limn→∞ an = limn→∞ cn =
L. Then limn→∞ bn = L.5
6
Prop: If (an )∞
n=1 converges, then the set { an |n ∈ N } is bounded.
Thm: If limn→∞ an = L, limn→∞ bn = M , and α ∈ R, then
(i) limn→∞ an + bn = L + M .
(ii) limn→∞ αan = αL.
(iii) limn→∞ an bn = LM .
L
if M 6= 0.7
(iv) limn→∞ abnn = M
Thm: For a sequence an ≥ 0, we have limn→∞ an = +∞ if an only if limn→∞ a1n = 0.8
Thm: “Least Upper Bound Principle” Every nonempty subset S of R that is bounded above has a supremum. Similarly,
every nonempty subset S of R that is bounded below has an infimum.9
Thm: “Monotone Convergence Theorem” A monotone increasing sequence that is bounded above converges. A monotone
decreasing sequence that is bounded below converges.10
Thm: Let (an ) be a sequence.
(i) If lim an is defined, then lim inf an = lim sup an = lim an .
(ii) if lim inf an = lim sup an = L, then lim an = L.11
∞
12
Def: A subsequence (an )∞
n=1 is a new sequence (ank )k=1 = (an1 , an2 , . . . ) where n1 < n2 < · · · .
13
Thm: If the sequence (an ) converges, then every subsequence converges to the same limit.
Thm: Every sequence (an ) has a monotonic subsequence.14
Cor: Let (an ) be a sequence. There exists a monotonic subsequence whose limit is lim sup an and there exists a
monotonic subsequence shows limit is lim inf an .15
Def: Let (an ) be a sequence in R. A subsequential limit is any real number (or symbol +∞, −∞) that is the limit of
some subsequence (ank ).16
Thm: Let (an ) be any sequence in R, and let S denote the set of subsequential limits of (an ).
(i) S is nonempty.
(ii) sup S = lim sup an and inf S = lim inf an .
(iii) lim an exists if an only if S has a single element, namely lim an .17
Thm: Let S denote the set os subsequential limits of a sequence (an ). Suppose (bn ) is a sequence in S ∩ R and that
t = lim tn . Then t ∈ S.18
Lma: “Nested Intervals Lemma” Suppose that In = [an , bn ] = { x ∈ R | an ≤ x ≤ bn } are nonempty closed intervals
such that In+1 ⊆ In for each n ≥ 1. Then the intersection ∩n≥1 In is nonempty.19
Thm: “Bolzano-Weierstrass Theorem” Every bounded sequence of real numbers has a convergent subsequence.20
Fact: For a bounded sequence (an ), lim sup an (lim inf an ) is the largest (smallest) possible value for a convergent
subsequence.21
Def: A sequence (an )∞
n=1 is called a Cauchy sequence if for every ε > 0, there is an integer N such that |am − an | < ε
for all m, n ≥ N .22
Def: A subset S of R is said to be complete if every Cauchy sequence converges to a point in S.23
Thm: “Completeness Theorem” A sequence of real numbers converges if and only if it is a Cauchy sequence. In particular,
R is complete.24
2
Series
P∞
Pn
Def: Given a sequence (an )∞
n=1 , the infinite series
n=1 an = limn→∞
k=1 ak converges if the limit exists, diverges
otherwise.25
Fact: Some series can be solved using a telescoping
sum, by cancelling elements between sequence terms.26
P
Thm: “nth term test” If lim an 6= 0, then
an diverges.27
Mike Janssen
3
TOPOLOGY OF RN
Analysis Study Guide
P∞
Thm: “Cauchy Criterion for Series” The following are equivalent for a series n=1 an .
(i) The series converges.
P∞
(ii) For every ε > 0, there is an N ∈ N so that k=n+1 ak < ε for all n ≥ N .
28
Pm
(iii) For every ε > 0, there is an N
k=n+1 ak < ε if n, m ≥ N .
Pn∈ N so that
Prop: If ak ≥ 0 for k ≥ 1 and sn = k=1 ak , then
P∞either
(i) (sn )∞
n=1 is bounded above, in which case
P∞ n=1 an converges.
29
(ii) (sn )∞
n=1 is unbounded, in which case
n=1 an diverges.
∞
Def: A sequence (an )n=0 is a geometric series with ratio r ifP
an+1 = ran for all n ≥ 0, or equivalently an = a0 rn .
∞
a
if |r| < 1.30
Thm: “Convergence of Geometric Series” A geometric series n=0 arn converges to 1−r
a
∞
Def: A sequence (an )n=0 is a p-series withP
power p if an = np for some a, p ∈ R.
∞
Thm: “Convergence of p-series” A p-series n=1 nap converges if and only if p > 1.31
Thm: “Comparison Test”
sequences (an ), (bn ) with |an | ≤ bn for all n ≥ 1. If (bn ) is summable, then (an )
P∞ ConsiderPtwo
∞
32
is summable and | n=1 aP
n| ≤
n=1 bn . If (an ) is not summable, then (bn ) is not summable.
Thm: “Ratio Test” A series
an of nonzero terms
(i) Converges absolutely if lim sup |an+1 /an | < 1,
(ii) Diverges if lim inf |an+1 /an | > 1.33
P∞
√
Thm: “Root Test” Suppose that an ≥ 0 for all n and let ` = lim sup n an . If ` < 1, then n=1 an converges absolutely,
and if ` > 1, the series diverges.34
Rb
P∞
Thm: “Integral Test” If f : R → R is positive and decreasing, then n=A f (n) converges if and only if limb→∞ A f (x)dx
exists (and is finite).35
P∞
Thm: “Limit Comparison Test” If n=1 bn is a convergent series of positive numbers bn and lim sup |abnn | < ∞ then
P∞
36
n=1 an converges.
Def: A sequence is alternating if it has the form (−1)n an or (−1)n+1 an where an ≥ 0 for all n ≥ 1.37
Lma: For x > −1, m ∈ N, (1 + x)m > 1 + mx.38
∞
Thm: “Leibniz Alternating Series Test” Suppose that (an )P
n=1 is a monotone decreasing sequence a1 ≥ a2 ≥ a3 ≥ · · · ≥ 0
∞
and that limn→∞ an = 0. Then the alternating series n=1 (−1)n an converges.39
Cor: Suppose that (an )∞
positive sequence and that limn→∞ an = 0. Then the difference
n=1 is a monotone decreasing
P∞
n
between the sum
of
the
alternating
series
(−1)
an and the NP
th partial sum is at most |aN |.
n=1
P∞
∞
Def: A series n=1 an is called absolutely convergent if the series n=1 |an | converges. A series that converges but
40
is not absolutely convergent is called
P∞ conditionally convergent.
Def: A rearrangement of a series n=1 an is another series with the same termsPin a different order. This can be
∞
described by a permutation π of the natural numbers N determining the series n=1 aπ(n) .41
Thm: For P
an absolutely convergent series, every rearrangement converges to the same limit.
∞
Lma: Let P n=1 an be a convergent series. Denote the positive terms
. . and the otherPterms as c1 ,P
c2 , . . . .
P∞ as b1 , b2 , .P
∞
∞
∞
∞
(i)
If
a
is
absolutely
convergent,
then
so
are
both
b
and
|c
|,
and
a
=
n
n
n
n
n=1
n=1
n=1
n=1
n=1 bn −
P∞
|cn |.
n=1 P
P∞
P∞
∞
(ii) If n=1 an is conditionally convergent,
then n=1 bn and n=1 |cn | both diverge.42
P∞
Thm: “Rearrangement Theorem” If n=1 is a conditionally convergent series, then for every real number L, there is a
rearrangement that converges to L.43
Pn
Lma: “Summation
by Parts
of real numbers. Define Xn = k=1 xk and
Pn
PmLemma” Suppose
Pm (xn ) and (yn ) are sequences
Yn = k=1 yn . Then n=1 xn Yn + n=1 Xn yn+1 = Xm Ym+1 .44
Thm: “Dirichlet’s Test” Suppose that (an )∞
bounded partial sums. If (bn )∞
n=1 is a sequence of real numbers with
n=1 is a
P∞
45
sequence of postive numbers decreasing
monotonically
to
0,
then
the
series
a
b
converges.
n
n
n=1
P∞
P∞
Thm: “Abel’s Test” Suppose that n=1 an converges and (bn ) is a monotonic convergent sequence. Then, n=1 an bn
converges.46
3
Topology of Rn
P∞
Def: The dot product or inner product of two vectors x, y ∈ Rn is hx, yi = n=1 xi yi .47
Thm: “Schwarz Inequality” For all x, y ∈ Rn , | hx, yi | ≤ ||x|| · ||y||. Equality holds if and only if x and y are colinear.48
Thm: “Triangle Inequality” The triangle inequality holds for the Euclidean norm on Rn : ||x + y|| ≤ ||x|| + ||y|| for all
x, y ∈ Rn . Moreover, equality holds if and only if either x = 0 or y = cx with c ≥ 0.49
Mike Janssen
4
FUNCTIONS
Analysis Study Guide
Def: A set V = {x1 , . . . , xm } ⊆ Rn is orthonormal if hvi , vj i = 0 when i 6= j and hvi , vi i = 1. If m = 0, then V spans
Rn and is called an orthonormal basis.50
Pm
Pm
2 1/2 51
.
Lma: Let {v1 , . . . , vn } be an orthonormal set in Rn . Then || i=1 ai xi || =
i=1 |ai |
Def: A sequence of points (xk ) in Rn converges to a point a if for every ε > 0, there is an integer N so that ||xk − a|| < ε
for all k > N . In this case, write limn→∞ xk = a.52
Lma: Let (xk ) be a sequence in Rn . Then limk→∞ xk = a if and only if limk→∞ ||xk − a|| = 0.53
Lma: A sequence xk = (xk,1 , . . . , xk,n ) in Rn converges to a point a = (a1 , . . . , an ) if and only if each coordinate
converges: limk→∞ xk,i = ai for 1 ≤ i ≤ n.54
Def: A sequence xk in Rn is Cauchy if for every ε > 0, there is an integer N so that ||xk − x` || < ε for all k, ` > N . A
set S ⊆ Rn is complete if every Cauchy sequence of points in S converges to a point in S.55
Thm: “Completeness Theorem for Rn ” Every Cauchy sequence in Rn converges. Thus, Rn is complete.56
Def: A point x is a limit point of a subset A ⊆ Rn if there is a sequence (ak )∞
k=1 with ak ∈ A such that x = limk→∞ ak .
A set A ⊆ Rn is closed if it contains all of its limit points.57
Def: A point x is a cluster point of a subset A ⊆ Rn if there is a sequence (an )∞
n=1 with an ∈ A \ {x} such that
x = limn→∞ an .58
Prop: If A, B ⊆ Rn are closed, then A ∪ B is closed. If { Ai | i ∈ I } is a family of closed subsets of Rn , then ∩i∈I Ai is
closed.59
E.g.: Let An = B n−1 (0). The family of sets {An }n∈N has every set closed, but ∪n∈N An = B1 (0), which is not closed.60
n
Def: If A is a subset of Rn , the closure of A is the set A consisting of all limit points of A.61
Def: The ball about a in Rn of radius r is the set Br (a) = { x ∈ Rn | ||x − a|| < r }. A subset U ⊆ Rn is open if for
every a ∈ U , there is some r > 0 so that the ball Br (a) is contained in U .62
Prop: Let A ⊆ Rn . Then A is the smallest closed set containing A. In particular, A = A.63
Thm: “Duality of Open and Closed Sets” A set A ⊆ Rn is open if and only if the complement of A, A0 = { x ∈ Rn | x ∈
/
A } is closed.64
–HERE MARKS THE END OF EXAM 1 MATERIAL–
Prop: If U and V are open subsets of Rn then U ∩ V is an open subset of Rn . If { Ui | i ∈ I } s a family of open subsets
of Rn , then ∪i∈I Ui is open.65
E.g.: Let An = B n+1 (0). The family of sets {An }n∈N has every set open, but ∩n∈N An = B1 (0), which is closed.66
n
Def: The interior, int X, of a set X is the largest open set contained in X. If int X = ∅, then X has empty interior.67
∞
Def: A subset A ⊆ Rn is compact if every sequence (ak )∞
k=1 of points in A has a convergent subsequence (aki )i=1 with
68
limit a = limi→∞ aki in A.
Def: A subset S of Rn is called bounded provided that there is a real number R such that S is contained in the ball
BR (0).69
Lma: A compact subset of Rn is closed and bounded.70
Lma: If C is a closed subset of a compact subset of Rn , then C is compact.71
Lma: The cube [a, b]n is a compact subset of Rn .72
Thm: “Heine-Borel Theorem” A subset of Rn is compact if and only if it is closed and bounded.73
Thm: “Cantor’s Intersection Theorem” If A1 ⊇ A2 ⊇ A3 ⊇ · · · is a decreasing sequence of nonempty compact subsets
of Rn , then ∩k≥1 Ak is not empty.74
Def: A set whose closure has no interior is nowhere dense. A point x of a set A is isolated if there is an ε > 0 such
that the ball Bε (x) intersects A only in the singleton {x}. A set A is perfect if each point x ∈ A is the limit of some
sequence in A \ {x}.75
4
Functions
Def: Let S ⊆ Rn and let f : S → Rm . If a ∈ S is a cluster point (limit point of S \ {a}) then a point v ∈ Rn is a
limit of f at a if for every ε > 0 there is an r > 0 so that ||f (x) − v|| < ε whenever 0 < ||x − a|| < r and x ∈ S.
Write limx→a f (x) = v.76
Def: Let S ⊆ Rn and let f : S → Rm . f is continuous at a ∈ S if for every ε > 0, there is an r > 0 such that, for
all x ∈ S with ||x − a|| < r, we have ||f (x) − f (a)|| < ε. Moreover, f is continuous on S if it is continuous at each
point a ∈ S. If f is not continuous at a, f is discontinuous at a.77
Mike Janssen
4
FUNCTIONS
Analysis Study Guide
Def: A function f : S → Rm is called Lipschitz if there is a constant C such that ||f (x) − f (a)|| ≤ C ||x − a|| for all
x, a ∈ S. The Lipschitz constant of f is the smallest choice of C for this condition to hold.78
Def: A function f : [a, b] → R satisfies a Lipschitz condition of order α > 0 if there is some positive constant M so that
|f (x1 ) − f (x2 )| ≤ M |x1 − x2 |α .79
Prop: Every Lipschitz function is continuous. 80
Cor: Every linear transformation A from Rn to Rm is Lipschitz, and therefore continuous.81
Def: The coordinate functions are, πj : Rn → R is given as πj (x1 , . . . , xn ) = xj for 1 ≤ j ≤ n, and i (t) = tei .82
Def: A function f : Rn → Rm has a removable singularity at a if limx→a f (x) exists, but does not equal f (a).83
Def: The limit of f as x approaches a from the right is L if for every ε > 0, there is an δ > 0 so that |f (x) − L| < ε for
all a < x < a + δ, written limx→a+ f (x) = L. Similarly, define the limit from the left, limx→a− f (x) = L.84
Def: When a function f on R has limits from the left and right that are different, we say f has a jump discontinuity.
A function on an interval is called piecewise continuous if on every finite subinterval, it has only a finite number of
jump discontinuities and is continuous at all other ponts.85
E.g.: The Heaviside function H(x) is defined to be 0 for all x < 0 and 1 for x ≥ 0.86
E.g.: The ceiling function dxe has countably many jump discontinuities.87
E.g.: For any subset A ⊆ Rn , the characteristic function of A is χA (x), which equals 1 if x ∈ A, and 0 otherwise.88
E.g.: Let f (x) = 0 when x ∈
/ Q and then f (x) = 1q when x = pq , in reduced form. Then, f (x) is continuous only at
irrational points, and discontinuous on the rationals.89
Def: Say that the limit of a function f (x) has x approaches a is +∞ if for every positive integer N , there is an r > 0
so that f (x) > N for all 0 < |x − a| < r. We write limx→a f (x) = +∞. We define the limit limx→a f (x) = −∞
similarly.90
Def: A function f is asymptotic to the curve g if limx→∞ |f (x) − g(x)| = 0.91
Def: A subset V ⊆ S ⊆ Rn is open in S (or relatively open) if there is an open set U in Rn such that U ∩ S = V .92
Thm: For a function f mapping S ⊆ Rn into Rm , the following are equivalent:
(i) f is continuous on S.
(ii) “Sequential characterization of continuity” For every convergent sequence (xn )∞
n=1 with limn→∞ xn = a ∈ S.
limn→∞ f (xn ) = f (a).
(iii) “Topological characterization of continuity” For every open set U in Rm , the set f −1 (U ) = { x ∈ S | f (x) ∈ U }
is open in S.93
Thm: If f, g are functions from a common domain S into Rm , a ∈ S such that limx→a f (a) = u and limx→a g(x) = v,
then
(i) limx→a f (x) + g(x) = u + v.
(ii) limx→a αf (x) = αu.
When the range is contained in R, say limx→a f (x) = u and limx→a g(x) = v, then
(iii) limx→a f (x)g(x) = uv, and
(iv) limx→a f (x)/g(x) = u/v provided v 6= 0.94
Thm: If f, g are functions form S to Rm that are continuous at a ∈ S and α ∈ R, then
(i) f + g is continuous at a.
(ii) αf is continuous at a.
When the range is contained in R,
(iii) f g is continuous at a
(iv) f /g is continuous at a provided that g(a) 6= 0.95
Def: A function f is a rational function if f (x) = p(x)/q(x) where p, q ∈ R[x] and q 6= 0. Rational functions are
continuous except where q(x) = 0.96
Def: If a function f : S → T and g : T → Rm , then the composition of g and f , denoted g ◦ f is the function that sends
x to g(f (x)).97
Thm: Suppose f : S → T and g : T → Rm . If f is continuous at a ∈ S and g is continuous at f (a) ∈ T , then g ◦ f is
continuous at a.98
Thm: Let C be a compact subset of Rn , and let f be a continuous function from C into Rm . Then the image set f (C)
is compact.99
Thm: “Extreme Value Theorem” Let C be a compact subset of Rn and let f be a continuous function from C into R. Then
there are points a and b in C attaining the minimum and maximum values of f on C. That is, f (a) ≤ f (x) ≤ f (b)
for all x ∈ C.100
Mike Janssen
5
INTRO TO CALCULUS
Analysis Study Guide
Def: A function f : R → Rm is a periodic function if there exists a d > 0 such that f (x) = f (x + d) for all x ∈ R. The
least such d is called the period of f . Then, f is d-periodic.101
Def: A function f : S → Rm is uniformly continuous if for every ε > 0 there is a δ > 0 so that ||f (x) − f (a)|| < ε
whenever ||x − a|| < δ for x, a ∈ S.102
Prop: Every Lipschitz function is uniformly continuous.103
Cor: Every linear transformation from Rn to Rm is uniformly continuous.104
Cor: Let f be a differentiable real-valued function on [a, b] with a bounded derivative; that is, there is M > 0 so that
|f 0 (x)| ≤ M for all a ≤ x ≤ b. Then f is uniformly continuous on [a, b].105
Thm: Suppose that C ⊆ Rn is conpact and f : C → Rn is continuous. Then f is uniformly continuous on C.106
Thm: “Intermediate Value Theorem” If f is a continuous real-valued function on [a, b] with f (a) < 0 < f (b), then there
exists a point c ∈ (a, b) such that f (c) = 0.107
Cor: If f is a continuous real-valued function on [a, b], then f ([a, b]) is a closed interval.108
Def: A path in S ⊆ Rn from a to b, both poionts in S, is the image of a continuous function γ from [0, 1] into S such
that γ(0) = a and γ(1) = b.109
Cor: Suppose that S ⊆ Rn and f is a continuous real-valued function on S. If there is a path from a to b in S and
f (a) < 0 < f (b), then there is a point c on the path so that f (c) = 0.110
Def: A function f is increasing on an interval (a, b) if f (x) ≤ f (y) whenever a < x ≤ y < b. It is strictly increasing on
(a, b) if f (x) < f (y) whenever a < x < y < b. Similarly, define decreasing and strictly decreasing functions. All of
these functions are called monotone.111
Prop: If f is an increasing function on the inerval (a, b), then the one-sided limits of f exist at each point c ∈ (a, b) and
limx→c− f (x) = L ≤ f (x) ≤ limx→c+ f (x) = M . For decreasing functions, the inequalities are reversed.112
Cor: The only type of discontinuity that a monotone function on an interval can have is a jump discontinuity.113
Cor: If f is a monotone function on [a, b] and the range of f intersects every nonempty open interval in [f (a), f (b)] then
f is continuous.114
Thm: A monotone function on [a, b] has at most countably many discontinuities.115
Thm: Let f be a continuous strictly increasing function on [a, b]. Then f maps [a, b] one-to-one and onto [f (a), f (b)].
Moreover the inverse function f −1 is also continuous and strictly increasing.116
Prop: Let f : S → Rm , S ⊆ Rn be a continuous function. If T ⊂ S is compact, f (T ) is compact.117
E.g.: The Cantor function c : [0, 1] → [0, 1] is an onto function defined such that c is constant over the intervals not in
the Cantor set, but is strictly increasing over the Cantor set. Also, c(C) = [0, 1].118
5
Intro to Calculus
(x0 )
exists. We write f 0 (x0 )
Def: A funciton f : (a, b) → R is differentiable at a point x0 ∈ (a, b) if limh→0 f (x0 +h)−f
h
119
for this limit.
Def: When f is differentiable at x0 , we define the tangent line to f at x0 to be the linear function T (x) = f (x0 ) +
f 0 (x0 )(x − x0 ).120
Prop: If f is differentiable at x0 , then it is continuous at x0 . Differentiable functions are continuous.121
Lma: Let f be a function on [a, b] that is differentiable at x0 . Let T (x) be the tangent line to f at x0 . Then T is the
(x)
unique linear function with the property limx→x0 f (x)−T
= 0.122
x−x0
Cor: If f (x) is a function on (a, b) and x0 ∈ (a, b), then the following are equivalent:
(i) f is differentiable at x0 .
(ii) There is a linear function T (x) and a function ε(x) on (a, b) such that limx→x0 ε(x) = 0 and f (x) = T (x) +
ε(x)(x − x0 ).
(iii) There is a function ϕ(x) on (a, b) such that f (x) = f (x0 ) + ϕ(x)(x − x0 ) and limx→x0 ϕ(x) exists.123
Thm: “Arithmetic of Derivatives” Let f and g be differentiable functions at the point a. Each of the functions cf (c a
constant), f + g, f g, and f /g are differentiable at a, except f /g if g(a) = 0. The formulas are124
(i) (cf )0 (a) = c · f 0 (a)
(ii) (f + g)0 (a) = f 0 (a) + g 0 (a)
0
(iii) “Product rule” (f g) (a) = f (a)g 0 (a) + f 0 (a)g(a)
0
(a)g 0 (a)
if g(a) 6= 0.
(iv) “Quotient rule” (f /g)0 (a) = g(a)f (a)−f
g 2 (a)
Thm: “The Chain Rule” Suppose that f is defined on [a, b] and has range contained in [c, d]. Let g be defined on [c, d].
Suppose that f is differentiable at x0 ∈ [a, b] and g is differentiable at f (x0 ). Then the composition h(x) = g(f (x))
is defined, and h0 (x0 ) = g 0 (f (x0 ))f 0 (x0 ).125
Mike Janssen
6
INTEGRATION
Analysis Study Guide
E.g.: The class of functions f (x) = xa sin(1/x) for x > 0 and f (0) = 0 for a > 0 is differentiable on [0, ∞), but the
derivative function is not continuous at 0.126
Def: A function f (x) is even if f (−x) = f (x) and odd when f (−x) = −f (x).127
–HERE MARKS THE END OF EXAM 2 MATERIAL–
Thm: “Fermat’s Thoerem” Let f be a continuous function on an interval [a, b] that takes its maximum or minimum
value at a point x0 . Then, exactly one of the following holds:
(i) x0 is an endpoint a or b,
(ii) f is not differentiable at x0 ,
(iii) f is differentiable at x0 and 0 f (x0 ) = 0.128
Thm: “Rolle’s Theorem” Suppose that f is a function that is continuous on [a, b] and differentiable on (a, b) such that
f (a) = f (b) = 0. THen there is a point c ∈ (a, b) such that f 0 (c) = 0.129
Thm: “Mean Value Theorem” Suppose that f is a function that is continuous on [a, b] and differentiable on (a, b). Then
(a) 130
.
there is a point c ∈ (a, b) such that f 0 (C) = f (b)−f
b−a
131
Cor: Let f be a differentiable function on [a, b].
(i) If f 0 (x) is (strictly) positive, then f is (strictly) increasing.
(ii) If f 0 (x) is (strictly) negative, then f is (strictly) decreasing.132
(iii) If f 0 (x) = 0 at every x ∈ (a, b), then f (x) is constant.133 (iv) If g is differentiable on [a, b] with g 0 (x) = f 0 (x),
then there is a constant c such that f (x) = g(x) + c.134
Def: If a twice differentiable function f has f 00 (x) positive on an interval [a, b] then f is convex or concave up. If f 00 (x)
is negative, then f is concave or concave down. The points where f 00 (x) changes sign are called inflection points.135
Thm: “Darboux’s Theorem/Intermediate Value Theorem for Derivatives” If f is differentiable on [a, b] and f 0 (a) < L <
f 0 (b), then there is a point x0 in (a, b) at which f 0 (x0 ) = L.136
Thm: Let f is a one-to-one continuous function on an open interval I, and let J = f (I). If f is differentiable at x0 ∈ I
1
and if f 0 (x0 ) 6= 0, then f −1 is differentiable at y0 = f (x0 ) and (f −1 )0 (y0 ) = f 0 (x
.137
0)
6
Integration
Def: Let f : [a, b] → R be a bounded function.138
(i) A partition of [a, b] is a finite set P = {a = x0 < x1 < · · · < xn−1 < xn = b}. Set ∆j = xj − xj−1 and define the
mesh of a partition P as mesh(P ) = max1≤j≤n ∆j .139
(ii) Let the maximum and minimum be Mj (f, P ) = sup{ f (x) | xj−1 ≤ x ≤ xj } and mj (f, P ) = inf{ f (x) | xj−1 ≤
x ≤ xj }.
Pn
(iii) Let the upper (Darboux) sum and lower (Darboux) sum be U (f, P ) =
j=1 Mj (f, P )∆j and L(f, P ) =
Pn
m
(f,
P
)∆
.
j
j
j=1
(iv) If given an evaluation sequence X = {x0j | 1 ≤ j ≤ n} with x0j ∈ [xj−1 , xj ] the Riemann sum is I(f, P, X) =
Pn
0
140
j=1 f (xj )∆j .
(v) A partition R is a refinement of a parition P provided P ⊆ R. If P and Q are partitions, then R is a common
refinement if P ∪ Q ⊆ R.
Lma: If R is a refinement of P , then L(f, P ) ≤ L(f, R) ≤ U (f, R) ≤ U (f, P ).141
Cor: If P and Q are any two partitions of [a, b], L(f, P ) ≤ U (f, Q).142
Def: Define the lower Darboux integral, L(f ) = supP L(f, P ) and the upper darboux integral, U (f ) = inf P U (f, P ). Note
that L(f ) ≤ U (f ).143 A bounded function f on a finite interval [a, b] is called Riemann integrable if L(f ) = U (f ).
Rb
In this case, we write L(f ) = a f (x)dx = U (f ).144
Thm: “Riemann’s Condition” Let f (x) : [a, b] → R be a bounded function. The following are equivalent:
(i) f is Riemann integrable.
(ii) For each ε > 0, there is a partition P so that U (f, P ) − L(f, P ) < ε.145
Cor: Let f be a bounded real-valued function on [a, b]. If there is a sequence of partitions of [a, b], Pn so that
limn→∞ U (f, Pn ) − L(f, Pn ) = 0, then f is Riemann integrable. Moreover, if Xn is any choice of points x0n,j
Rb
selected from each interval of Pn , then limn→∞ I(f, Pn , Xn ) = a f (x)dx.
Thm: Let f (x) : [a, b] → R be a bounded function. The following are equivalent:146
(i) f is Riemann integrable.
(ii) For each ε > 0, there is a partition P so that U (f, P ) − L(f, P ) < ε.
(iii) “Cauchy Criterion for Integrability” For every ε > 0, there is a δ > 0 so that every partition Q such that
mesh(Q) < δ satisfies U (f, Q) − L(f, Q) < ε.147
Mike Janssen
7
RIEMANN-STIELTJES INTEGRATION
Analysis
Study Guide
Rb
Thm: Let f (x) : [a, b] → R be a bounded function, then f is Riemann integrable with a f (x)dx = L if and only if for
every ε > 0, there is a δ > 0 so that every partition Q such that mesh(Q) < δ and every choice of evaluation sequence
X satisfies |I(f, Q, X) − L| < ε.148
Thm: Every monotone function on [a, b] is Riemann integrable.149
Thm: Every continuous function on [a, b] is integrable.150
R∞
Rb
Def: Say that f is Riemann integrable on [a, ∞) if the improper integral a f (x)dx := limb→∞ a f (x)dx exists.151
Thm: “Arithmetic of Integrals” Let f and g be integrable function on [a, b] and c ∈ R. Then
Rb
Rb
Rb
Rb
(i) cf is integrable and a cf (x)dx = c a f (x)dx.
(ii) f + g is integrable and a (f + g)(x)dx = a f (x)dx +
Rb
g(x)dx.152
a
Rb
Rb
Thm: If f and g are integrable on [a, b] and if f (x) ≤ g(x) for all x ∈ [a, b], then a f (x)dx ≤ a g(x)dx.153
Rb
Rb
Thm: If f is integrable on [a, b], then |f | is integrable on [a, b] and | a f (x)dx| ≤ a |f (x)|dx.154
Rb
Thm: If f : [a, b] → R with c ∈ (a, b) and f is integrable on [a, c] and [c, b] then f is integrable on [a, b] with a f (x) =
Rc
Rb
f (x)dx + c f (x)dx.155
a
Thm: “Fundamental Theorem of Calculus
I” Let f be a bounded Riemann integrable function on [a, b] and let F :
Rx
[a, b] → R be defined as F (x) = a f (t)dt. Then F is continuous, and if f is continuous at a point x0 , then F is
differentiable at x0 with F 0 (x0 ) = f (x).156
Def: A function f on [a, b] has an antiderivative if there is a continuous function F : [a, b] → R such that F 0 (x) = f (x)
for all x ∈ [a, b].157
Cor: “Fundamental Theorem of Calculus II” Let f be a continuous function on [a, b]. Then f has an antiderivative.
Rb
Moreover, if G is any antiderivative of f , then a f (x)dx = G(b) − G(a).158
Rb
Lma: Suppose that f : [a, b] → R is an integrable function bounded by M : [a, b] → R. Then | a f (t)dt| ≤ M (b − a).159
Thm: “Integration by Parts” GIven two differentiable functions f : [a, b] → R and g : [a, b] → R, the integral
Rb 0
Rb
f (x)g(x)dx = f (x)g(x)|ba − a f (x)g 0 (x)dx.160
a
Thm: “Substitution/Change of Variable” Let u be a C 1 function on [a, b] and let f be continuous on [c, d] containing
Rb
R u(b)
the range of u. Then a f (u(x))u0 (x)dx = u(a) f (t)dt.161
Thm: “Intermediate/Mean Value Theorem for Integrals” If f is a continuous function on [a, b], then there is a point
Rb
1
f (x)dx = f (c).162
c ∈ (a, b) such that b−a
a
7
Riemann-Stieltjes Integration
Def: Consider bounded functions f, g : [a, b] → R. Given a partition of [a, b], P = {a = x0 < x1 < · · · < xn = b},
and an evaluation sequence X = (x01 , x02 , · · · , x0n ) for P , we define the Riemann-Stieltjes sum, or briefly, the R-S
sum, for f w.r.t. g using P and X, as
Ig (f, P, X) =
n
X
f (x0j )[g(xj ) − g(xj−1 )].
j=1
We say f is Riemann-Stieltjes integrable with respect to g, denoted f ∈ R(g), if there is a number L so that for
all ε > 0, there is a partition Pε so that, for all partitions P with P ⊇ Pε and all evaluation sequences X for P , we
have
|Ig (f, P, X) − L| < ε.
Thm: If f1 ∈ R(g) and f2 ∈ R(g) on [a, b], and c1 , c2 ∈ R, then c1 f1 + c2 f2 ∈ R(g) on [a, b] and
Z
b
Z
c1 f1 + c2 f2 dg = c1
b
Z
f1 dg + c2
a
a
b
f2 dg.
a
If f ∈ R(g1 ) and f ∈ R(g2 ) on [a, b], then f ∈ R(g1 + g2 ) on [a, b] and
Z
b
Z
f d(g1 + g2 ) =
a
b
Z
f dg1 +
a
Mike Janssen
b
f dg2 .
a
7
RIEMANN-STIELTJES INTEGRATION
Analysis
Study Guide
Finally, if a < b < c and f is R-S integrable w.r.t. g on both [a, b] and [b, c], then it is R-S integrable w.r.t. g on
[a, c] and
Z c
Z b
Z c
f dg.
f dg +
f dg =
b
a
a
Thm: Suppose f, g : [a, b] → R are bounded and ϕ : [c, d] → [a, b] is a strictly increasing, continuous function onto [a, b].
Let F = f ◦ ϕ and G = g ◦ ϕ. If f ∈ R(g) on [a, b] then F ∈ R(G) on [c, d] and
Z
d
Z
F dG =
c
b
f dg.
a
Thm: Let f, g : [a, b] → R be bounded functions. If f ∈ R(g) on [a, b], then g ∈ R(f ) on [a, b] and
b
Z
Z
b
gdf = g(b)f (b) − g(a)f (a).
f dg +
a
a
Thm: If f : [a, b] → R has f ∈ R(g), where g : [a, b] → R is C 1 , then f g 0 is Riemann integrable on [a, b] and
Z
b
Z
f dg =
a
b
f (x)g 0 (x)dx.
a
Def: Let f and g be bounded functions on [a, b]. As usual, let P = {x0 < x1 < · · · < xn } be a partition of [a, b] and
recall that
f (x).
f (x) and mj (f, P ) =
inf
Mj (f, P ) =
sup
xj−1 ≤x≤xj
xj−1 ≤x≤xj
Define the upper sum with respect to g and the lower sum with respect to g to be
Ug (f, P ) =
Lg (f, P ) =
n
X
i=1
n
X
Mi (f, P )[g(xi ) − g(xi−1 )]
mi (f, P )[g(xi ) − g(xi−1 )].
i=1
Rmrk: If g is increasing then, for all evaluation sequences X, Lg (f, P ) ≤ Ig (f, P, X) ≤ Ug (f, P ).
Lma: (Refinement Lemma) Let f, g be bounded functions on [a, b] and P, R partitions of [a, b]. Assume that g is
increasing. If R is a refinement of P , then
Lg (f, P ) ≤ Lg (f, R) ≤ Ug (f, R) ≤ Ug (f, P ).
Cor: Let f, g be bounded functions on [a, b] and assume that g is increasing. If P and Q are any two partitions of [a, b],
Lg (f, P ) ≤ Ug (f, Q).
Def: We define Lg (f ) = supP Lg (f, P ) and Ug (f ) = inf P Ug (f, P ).
Prop: (Riemann-Stieltjes Condition) Let f, g be bounded functions on [a, b] and assume that g is increasing on [a, b].
TFAE:
1. f is R-S integrable w.r.t. g.
2. Ug (f ) = Lg (f ), and
3. for each ε > 0, there is a partition P so that Ug (f, P ) − Lg (f, P ) < ε.
Def: Given a function f : [a, b] → R and a partition [a, b], say P = {a = x0 < x1 < · · · < xn = b}, then the variation
of f over P is
n
X
V (f, P ) :=
|f (xi ) − f (xi−1 )|.
i=1
Mike Janssen
8
Analysis Study Guide
NORMED VECTOR SPACES
Rmrk: If P is a partition of [a, b] and Q is a partition of [b, c], then for a function f : [a, c] → R, V (f, P ∪ Q) =
V (f, P ) + V (f, Q).
Rmrk: If R is a refinement of P , then V (f, P ) ≤ V (f, R).
Def: The (total) variation of f on [a, b] is
Vab f = sup {V (f, P ) : P a partition of [a, b]}.
We say f is of bounded variation on [a, b] if Vab f is finite.
Lma: If a < b < c and f : [a, c] → R is given, then
Vac f = Vab f + Vbc f.
Thm: If f : [a, b] → R is of bounded variation on [a, b], then there are increasing functions g, h : [a, b] → R so that
f = g − h.
Thm: If f : [a, b] → R is bounded by M , g : [a, b] → R is of bounded variation and f ∈ R(g) on [a, b], then
Z
b
f dg ≤ M · Vab g.
a
Thm: If f : [a, b] → R is continuous and g : [a, b] → R is increasing, then f ∈ R(g) on [a, b].
Cor: If f : [a, b] → R is continuous and g : [a, b] → R is of bounded variation, then f ∈ R(g) on [a, b].
8
Normed Vector Spaces
Def: Let V be a vector space over R. A norm on V is a function ||·|| on V taking values in [0, +∞) with the
following properties:
1. (positive definite) ||x|| = 0 if and only if x = 0,
2. (homogeneous) ||αx|| = |α| ||x|| for all x ∈ V and α ∈ R, and
3. (triangle inequality) ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ V .
E.g.: Let V = Rn . Then the following are norms:
n
X
||x|| = ||(x1 , · · · , xn )||2 =
||x||1 = ||(x1 , · · · , xn )||1 =
!1/2
2
|xi |
i=1
n
X
|xi |
i=1
||x||∞ = ||(x1 , · · · , xn )||∞ = max |xi |.
1≤i≤n
E.g.: Fix a real number p in [1, ∞). Define the Lp [a, b] norm on C[a, b] by
Z
||f ||p =
!1/p
b
p
|f (x)| dx
.
a
Def: A complete normed vector space V is called a Banach space.
Prop: A sequence xn in a normed vector space V converges to a vector x if and only if for each open set U containing
x, there is an integer N so that xn ∈ U for all n ≥ N .
Def: A subset K of a normed vector space V is compact if every sequence (xn ) of points in K has a subsequence (xni )
which converges to a point in K.
Mike Janssen
8
NORMED VECTOR SPACES
8.1
Analysis Study Guide
8.1
Inner Product Spaces
Inner Product Spaces
Def: An inner product on a vector space V is a function hx, yi on pairs (x, y) of vectors in V × V taking values in
R satisfying the following properties:
1. (positive definiteness) hx, xi ≥ 0 for all x ∈ V and hx, xi = 0 only if x = 0.
2. (symmetry) hx, yi = hy, xi for all x, y ∈ V .
3. (bilinearity) For all x, y, z ∈ V and scalars α, β ∈ R, hαx + βy, zi = α hx, zi + β hy, zi.
Given an inner product space, it is easy to check that the following definition gives us a norm:
||x|| = hx, xi
1/2
.
E.g.: The space C[a, b] can be given an inner product
Z
hf, gi =
b
f (x)g(x)dx,
a
which gives rise to the L2 norm.
Thm: (Cauchy-Schwarz Inequality) For all vectors x, y in an inner product space V ,
| hx, yi | ≤ ||x|| ||y|| .
Equality holds if and only if x and y are colinear.
Cor: Let V be an inner product space with induced norm ||·||. Then the inner product is continuous (i.e., if xn converges
to x and yn converges to y, then hxn , yn i converges to hx, yi).
8.2
Orthonormal Sets
Def: Two vectors x and y are called orthogonal if hx, yi = 0. A collection of vectors {en : n ∈ S} in V is called
orthonormal if ||en || = 1 for all n ∈ S and hen , em i = 0 for n 6= m ∈ S. This set is called an orthonormal basis
if in addition this set is maximal with respect to being an orthonormal set.
Prop: An orthonormal set is linearly independent. An orthonormal basis for a finite-dimensional inner product space
is a basis.
√
√
Lma: The functions 1, 2 cos nθ, 2 sin nθ : n ≥ 1 form an orthonormal set in C[−π, π] with the inner product
Rπ
1
hf, gi = 2π
f (θ)g(θ)dθ.
−π
N
P
Def: A trigonometric polynomial is a finite sum f (θ) = A0 +
Ak cos kθ + Bk sin kθ.
∞
P
k=1
Rπ
1
f (t)dt, and
Def: Denote the Fourier series of f ∈ C[−π, π] by f = A0 +
An cos nθ + Bn sin nθ, where A0 = 2π
−π
n=1
R
R
π
π
for n ≥ 1, An = π1 −π f (t) cos ntdt and Bn = π1 −π f (t) sin ntdt.
Lma: Let {e1 , · · · , en } be an orthonormal set in an inner product space V . If M is the subspace spanned by {e1 , · · · , en },
n
P
then every vector x ∈ M can be written uniquely as
αi ei , where αi = hx, ei i. In other words, the set {e1 , · · · , en }
i=1
is linearly independent. Moreover, for each y ∈ v with hy, ei i = βi , and each x =
n
P
j=1
αj ej in M , hx, yi =
n
P
αi βi .
i=1
Cor: If V is an inner product space of finite dimension n, then it has an orthonormal basis {ei : 1 ≤ i ≤ n} and the
inner product is given by
* n
+
n
n
X
X
X
α i ei ,
βj e j =
αi βi .
i=1
j=1
Mike Janssen
i=1
8
Analysis Study Guide
NORMED VECTOR SPACES
8.3
8.3
Orthogonal Expansions in IPS
Orthogonal Expansions in IPS
Def: A projection is a linear map P such that P 2 = P . In addition, say that P is an orthogonal projection if
ker P is orthogonal to Ran P .
Rmrk: Observe that ker P = Ran (I − P ) because P x = 0 if and only if (I − P )x = x. Further, x = (I − P )y implies
P x = (P − P 2 )y = 0. Thus, when P is an orthogonal projection, the vectors P x and (I − P )x are orthogonal.
Thm: (Projection Theorem) Let A = {e1 , · · · , en } be an orthonormal set in an IPS V and let M be the subspace spanned
n
P
by A. Define P : V → M by P y =
hy, ej i ej , for each y ∈ V . Then P is the orthogonal projection onto M and
j=1
||y|| ≥
n
X
2
hy, ej i .
j=1
Moreover, for all v ∈ M ,
2
2
2
||y − v|| = ||y − P y|| + ||P y − v|| .
In particular, P y is the closest vector in M to y.
Prop: (Bessel’s Inequality) Let {en : n ∈ S} be an orthonormal set in an inner product space V . For each vector x ∈ V ,
P
2
| hx, en i |2 ≤ ||x|| .
n∈S
Def: A complete inner product space is called a Hilbert space.
E.g.: The space `2 consists of all sequences x = (xn )∞
n=1 such that ||x||2 =
`2 is given by hx, yi =
∞
P
∞
P
n=1
x2n
1/2
is finite. The inner product on
xn yn .
n=1
Thm: The space `2 is complete.
Proof. Forthcoming.
Thm: (Parseval’s Theorem) Let S ⊂ N and E = {en : n
H. Then the
P∈ S} be an orthonormal set in a Hilbert space
subspace M = span {E} consists of all vectors x =
αn en , where the coefficient sequence (αn )∞
belongs
to `2 .
n=1
n∈S
Further, if x is a vector in H, then x ∈ M if and only if
X
2
| hx, en i |2 = ||x|| .
n∈S
Cor: Let E = {en : n ∈ S} be an orthonormal set in a HilbertP
space H. Then there is a continuous linear orthogonal
projection PE of H onto M = span {E} given by PE (x) =
hx, en i en .
n∈S
Cor: If E = {ei : i ≥ 1} is an orthonormal basis for a Hilbert space H, every vector x ∈ H may be uniquely expressed
∞
P
as x =
αi ei , where αi = hx, ei i.
i=1
8.4
Finite-Dimensional Normed Spaces
Lma: If {v1 , · · · , vn } is a LI set in a normed vector space
n (V, ||·||),
then there exist positive constants 0 < c < C such
P
that for all a = (a1 , · · · , an ) ∈ Rn , we have c ||a||2 ≤ ai vi ≤ C ||a||2 .
i=1
Rmrk: This effectively means that every finite-dimensional normed vector space has the same topology as (Rn , ||·||2 ) in
the sense that they have the “same” convergent sequences, the same open/closed sets, etc.
Cor: A subset of a finite-dimensional normed vector space is compact if and only if it is closed and bounded.
Cor: A finite-dimensional subspace of a normed vector space is complete, and in particular it is closed.
Thm: Let (V, ||·||) be a normed vector space, and let W be a finite-dimensional subspace of V . Then for any v ∈ V ,
there is at least one closest point w∗ ∈ W so that ||v − w∗ || = inf {||v − w|| : w ∈ W }.
Mike Janssen
9
Analysis Study Guide
LIMITS OF FUNCTIONS
9
9.1
Limits of Functions
Limits of Functions
Def: Let (fn ) be a sequence of functions from S ⊂ Rn into Rm . This sequence converges pointwise to a function
f if
lim fn (x) = f (x) for all x ∈ S.
n→∞
Def: Let (fn ) be a sequence of functions from S ⊂ Rn to Rm . This sequence converges uniformly to a function
f if for every ε > 0, there is an integer N so that
||fn (x) − f (x)|| < ε for all s ∈ S and n ≥ N.
Thm: For a sequence of functions (fn ) in Cb (S, Rm ) (the space of all bounded continuous functions from S to Rm ), (fn )
converges uniformly to f if and only if
lim ||fn − f ||∞ = 0.
n→∞
n
E.g.: The sequence fn (x) = x for x ∈ [0, 1] converges pointwise to χ{1} . The functions fn are polynomials, and hence
not only continuous but even smooth; while the limit function has a discontinuity at the point 1. For each n ≥ 1,
we have fn (1) = 1 and so
fn − χ{1} = sup |xn − 0| = 1.
∞
0≤x<1
So fn does not converge in the uniform norm. Indeed, to contradict the definition, take ε = 1/2. For each n, let
xn = 2−1/2n . Then
1
|fn (xn ) − χ{1} (xn )| = √ > ε.
2
Hence no integer N satisfies the definition.
9.2
Uniform Convergence and Continuity
Thm: Let (fn ) be a sequence of continuous functions mapping a subset of S of Rn into Rm that converges uniformly
to a function f . Then f is continuous.
Thm: (Completeness of C(K)) If K is a compact set, the space C(K) of all continuous functions on K with the sup
norm is complete.
9.3
Uniform Convergence and Integration
Thm: (Integral Convergence Theorem) Let (fn ) be a sequence of continuous functions on the closed interval [a, b]
converging uniformly to f (x) and fix c ∈ [a, b]. Then the functions
Z x
Fn (x) =
fn (t)dt for n ≥ 1
c
Rx
converge uniformly on [a, b] to the function F (x) = c f (t)dt.
Cor: Suppose that (fn ) is a sequence of continuously differentiable functions on [a, b] such that (fn0 ) converges uniformly
to a function g and there is a point c ∈ [a, b] so that lim fn (c) = γ exists. Then (fn ) converges uniformly to a
n→∞
differentiable function f with f (c) = γ and f 0 = g.
Rd
Prop: Let f (x, t) be a continuous function on [a, b] × [c, d]. Define F (x) = c f (x, t)dt. Then F is continuous on [a, b.
∂
Thm: (Leibniz’s Rule) Suppose that f (x, t) and ∂x
f (x, t) are continuous functions on [a, b] × [c, d]. Then F (x) =
Rd
f (x, t)dt is differentiable and
c
Z d
∂
0
F (x) =
f (x, t)dt.
∂x
c
Mike Janssen
9
Analysis Study Guide
LIMITS OF FUNCTIONS
9.4
9.4
Series of Functions
Series of Functions
∞
P
Thm: Let (fn ) be a sequence of continuous functions from a subset S of Rn into Rm . If
fn (x) converges uniformly,
n=1
then it is continuous.
Def: Let S ⊂ Rn . We say that a series of functions fk from S to Rm is uniformly Cauchy on S if for every ε > 0
there is N so that
l
X
fi (x) ≤ ε whenever x ∈ S and l > k ≥ N.
i=k+1
Thm: A series of functions converges uniformly if and only if it is uniformly Cauchy.
Thm: (Weierstrass M-Test) Suppose that an (x) is a sequence of functions on S ⊂ Rk into Rm and (Mn ) is a sequence
of real numbers so that
||an ||∞ = sup ||an (x)|| ≤ Mn for all x ∈ S.
x∈S
If
∞
P
P
Mn converges, then the series
an (x) converges uniformly on S.
n=1
P −k
2 cos(10k πx) is continuous on R and differentiable at no point in R.
E.g.: The function f (x) =
n≥1
k≥1
9.5
Power Series
Def: A power series is a series of functions of the form
∞
X
an xn = a0 + a1 x + a2 x2 + · · · .
n=0
Thm: (Hadamard’s Theorem) Given a power series
∞
P
an xn , there is R ∈ [0, +∞) ∪ {+∞} so that the series converges
n=0
for all x with |x| < R and diverges for all x with |x| > R. Moreover, the series converges uniformly on each closed
interval [a, b] contained in (−R, R). Finally, if α = lim sup |an |1/n , then
n→∞


+∞
R= 0

1
if α = 0
if α = +∞
if α ∈ (0, +∞).
α
We call R the radius of convergence of the power series.
∞
∞
P
P
Thm: (Term-by-Term Differentiation of Series) If f (x) =
an xn has radius of convergence R > 0, then
nan xn−1
n=0
n=1
has radius of convergence R, f is differentiable on (−R, R) and, for x ∈ (−R, R),
0
f (x) =
∞
X
nan xn−1 .
n=1
Further,
∞
P
an n+1
x
has radius of convergence R and, for x ∈ (−R, R),
n
+1
n=0
Z
0
9.6
x
∞
X
an n+1
f (t)dt =
x
.
n
+1
n=0
Compactness and Subsets of C(K)
Def: A family of functions F mapping a set S ⊂ Rn into Rm is equicontinuous at a point a ∈ S if for every
ε > 0, there is an r > 0 such that
||f (x) − f (a)|| < ε whenever ||x − a|| < r and f ∈ F.
Mike Janssen
10
Analysis Study Guide
METRIC SPACES
The family F is equicontinuous on a set S if it is equicontinuous at every point in S. The family F is uniformly
equicontinuous on S if for each ε > 0, there is an r > 0 such that
||f (x) − f (y)|| < ε whenever
||x − y|| < r, x, y ∈ S and f ∈ F.
Lma: Let K be a compact subset of Rn . A compact subset F of C(K, Rm ) is equicontinuous.
Prop: If F is an equicontinuous family of functions on a compact set, then it is uniformly equicontinuous.
Def: A subset S of K is called an ε-net of K if
[
K⊂
Bε (a).
a∈S
A set K is totally bounded if it has a finite ε-net for every ε > 0.
Lma: Let K be a bounded subset of Rm . Then K is totally bounded, man.
Cor: Let K be a compact subset of Rm . Then K contains a sequence {xi : i ≥ 1} that is dense in K. Moreover, for
any ε > 0, there is an integer N so that {xi : 1 ≤ i ≤ N } forms an ε-net for K.
Thm: (Arzela-Ascoli) Let K be a compact subset of Rn . A subset F of C(K, Rm ) is compact if and only if it is closed,
bounded, and equicontinuous.
10
10.1
Metric Spaces
Definitions and Examples
Def: Let X be a set. A metric on a set X on a set X is a function d : X × X → [0, ∞) with the following properties:
(1) d(x, y) = 0 if and only if x = y,
(2) d(x, y) = d(y, x) for all x, y ∈ X,
(3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
A metric space is a set X with a metric d, denoted (X, d).
E.g.: If X is a subset of a normed space V , define d(x, y) = ||x − y||.
E.g.: Define a metric on Z by ρ2 (n, n) = 0 and ρ2 (m, n) = 2−d , where d is the largest power of 2 dividing m − n.
E.g.: If X is a closed subset of Rn , let K(X) denote the collection of all nonempty compact subsets of X. If A is a
compact subset of X and x ∈ X, we define dist(x, A) = inf ||x − a||. Then we define the Hausdorff metric on
a∈A
K(X) by
dH (A, B) = max sup dist(a, B), sup dist(b, A)
a∈A
b∈B
= max sup inf ||a − b|| , sup inf ||a − b|| .
a∈A b∈B
b∈B a∈A
Prop: When X ⊆ Rn is closed, then (K(X), dH ) is a complete metric space.
Thm: Let f : (X, ρ) → (Y, σ). TFAE:
1. f is continuous on X;
2. for every convergent sequence (xn )∞
n=1 with lim xn = a in X, we have lim f (xn ) = f (a); and
3. f −1 (U ) is open for every open set U ⊆ Y .
Thm: The space Cb (X) of all bounded continuous functions on X with the sup norm ||f || = sup {|f (x)| : x ∈ X} is
complete.
Rmrk: Given a metric space (X, d), (X, d) (the bounded metric space) has the same topology as (X, d).
Mike Janssen
DISCRETE DYNAMICAL SYSTEMSAnalysis
11
10.2
Study Guide
10.2
Compact Metric Spaces
Compact Metric Spaces
Thm: (Borel-Lebesgue) For a metric space X, TFAE:
1. X is compact.
2. Every collection of closed subsets of X with the FIP has nonempty intersection.
3. X is sequentially compact.
4. X is complete and totally bounded.
Prop: A metric space X is complete if and only if every decreasing sequence of nonempty closed sets (balls) with radii
going to zero has nonempty intersection.
11
11.1
Discrete Dynamical Systems
Fixed Points and the Contraction Principle
Def: For a metric space (X, d), call T : X → X a contraction if there is c ∈ (0, 1) such that d(T x, T y) ≤ cd(x, y)
for all x, y ∈ X. That is, T satisfies a Lipschitz condition with c < 1. For T : X → X call x0 ∈ X a fixed point for
T if T x0 = x0 .
E.g.: Let X = R, T (x) = mx + b (an affine map). If m = 0, b is a fixed point and imT = B. If b = 0 then T (x) = mx
and x = 0 is a fixed point. If m 6= 1, this is the only fixed point. If |m| < 1, then for any x ∈ T , T n (x) → x. If
|m| > 1, then T n (x) diverges. If m = −1, then T (T (x)) = T (−x) = x so for every x 6= 0, wehave a period 2 orbit.
For general b, we can see inductively that xn = mn x + b(1 + m + · · · + mn−1 ) = mn x + b
1−mn
1−m
for m 6= 1. If
b
1−m .
|m| < 1, then this converges to
If m > 1 and x 6= 0, this sequence diverges to ∞. To find fixed points, we have
x = T (x) = mx + b implies x = b/(1 − m) for m 6= 1. If |m| > 1 and xn = T (xn−1 ), then xn−1 = T −1 (xn ). Now
T −1 is an affine map with slope |1/m| < 1. Define x−n = (T −1 )n (x), which converges to b/(1 − m) as n → ∞. It
follows that T n (x) diverges for all x 6= b/(1 − m). It turns out that T is a contraction whenever |m| < 1. Notice we
have fixed points even for |m| ≥ 1.
Thm: (Banach Contraction Principle) Let (X, d) be a complete metric space. Let T : X → X be a contraction. Then T
has a unique fixed point x∗ , and for all x ∈ X, T n (x) → x∗ . Moreover for any x ∈ X,
d(x∗ , T n (x)) ≤ cn d(x∗ , x) ≤
11.2
cn
d(x, T (x)).
1−c
Iterated Function Systems
Def: We call a linear map T : Rn → Rn a similitude if there is r > 0 such that for all x, y ∈ Rn , ||T x − T y|| =
r ||x − y||. If r < 1, then T is a contraction, called a contractive similitude. An iterated function system
(IFS) is a finite set of contractive similitudes.
Fact: If A1 , · · · , Ar , B1 , · · · , Br ∈ K(X), then dH (A1 ∪ · · · ∪ Ar , B1 ∪ · · · ∪ Br ) ≤ max {dH (A1 , B1 ), · · · , dH (Ar , Br )}.
Thm: Let X ⊆ Rn be closed. Suppose {T1 , · · · , Tr } is an IFS on X where Ti has Lipschitz constant si . Define T on
K(X) by T (A) = T1 (A) ∪ · · · ∪ Tr (A), and let s = max {s1 , · · · , sr }. Then
1. T is a contraction on K(X) with Lipschitz constant s,
2. There is a unique A ∈ K(X) such that A = T (A), and
3. For any B ∈ K(X), dH (T k (B), A) ≤ sk dH (B, A) ≤ sk /(1 − s)dH (T (B), B).
E.g.: Concretely, how do we find the magical A for a given IFS {T1 , · · · , Tr }? Consider words w in {1, · · · , r} and let |w|
denote the length of w. Given a word i1 i2 · · · im = w, define a map Tw : X → X by Ti1 ◦ Ti2 ◦ · · · ◦ Tim . Notice that
Tw is a contractive similitude with constant si1 · · · sim ≤ s|w| (where the sij are the constants for Tij respectively
and s is their max). There are three approaches:
i
(1) Find B ∈ K(X) such that T (B) ⊆ B. Then A = ∩∞
k=1 T (B).
Mike Janssen
12
APPROXIMATION BY POLYNOMIALS
Analysis
Study Guide
(2) For each word w, let aw be the fixed point of Tw . Then A = {aw : w a word in 1, · · · , r}.
(3) Pick any point a ∈ A. Then A = {Tw (a) : w a word in 1, · · · , r}.
Thm: (Fixed points of IFS’s are dense in K(Rn )) For any C ⊆ Rn compact and ε > 0, there is an IFS {T1 , · · · , Tr } so
that its fixed point set A satisfies dH (A, C) < ε.
12
12.1
Approximation by Polynomials
Taylor Series
Def: For f : [a, b] → R, n times differentiable at c ∈ (a, b), the nth Taylor polynomial for f at c is
Pn (x) = f (c) + f 0 (c)(x − c) + f 00 (c)
(x − 2)2
(x − c)n
+ · · · + f (n) (c)
.
2!
n!
(k)
Lma: Pn is the unique polynomial of degree at most n such that Pn (c) = f (k) (c).
Thm: Suppose f is n + 1 times differentiable on [a, b]. Let x, c ∈ (a, b). There is d between x and c such that
f (x) − Pn (x) = f (n+1) (d)
(x − c)n+1
.
(n + 1)!
Furthermore, if |f (n+1) | is bounded by M on [a, b], then
|f (x) − Pn (x)| ≤ M
|x − c|n+1
.
(n + 1)!
Def: Formally, we say f is O(g) near a ∈ R if there is some M > 0, δ > 0 such that |f (x)| ≤ M |g(x)| for all x with
0 < |x − a| ≤ δ.
Prop: Rules for computation: O(f ) ± O(g) = O(max {f, g}), O(f )O(g) = O(f g).
E.g.: O((x − a)m ) ± O((x − a)n ) = O((x − a)min{m,n} ), and O((x − a)m )((x − a)n ) = O((x − a)m+n ).
Rmrk: Taylor’s Theorem says O(f − Pn ) = O((x − c)n+1 ).
12.2
How not to Approximate a Function
Thm: (Weierstrass Approximation Theorem) Let f be any continuous real-valued function on [a, b]. Then there is a
sequence of polynomials pn that converges uniformly to f on [a, b].
12.3
Bernstein’s Proof of the Weierstrass Theorem
Def: We define the Bernstein polynomials to be Pkn (x) =
(Bn f )(x) =
n
k
xk (1 − x)n−k . We define a polynomial Bn f by
n
n
X
X
k
k
n k
f
Pkn (x) =
f
x (1 − x)n−k .
n
n
k
k=0
k=0
Prop: The map Bn is linear and monotone. That is, for all f, g ∈ C[0, 1] and α ∈ R,
(1) Bn (f + g) = Bn f + Bn g.
(2) Bn (αf ) = αBn f
(3) Bn f ≥ 0 if f ≥ 0
(4) Bn f ≥ Bn g if f ≥ g
(5) |Bn f | ≤ Bn g if |f | ≤ g.
Mike Janssen
NOTES
Analysis Study Guide
66 D.S.
Notes
1 Donsig,
825 Notes, 08/29.
825 Notes, 08/29.
3 Donsig, 825 Notes, 08/31.
4 D&D Definition 2.3.1. p. 38.
5 D&D. Theorem 2.3.6. p. 39.
6 D&D. Proposition 2.4.2. p. 42; Ross, Theorem 9.1. p. 43.
7 D&D. Theorem 2.4.3. p. 42; Ross, Thm 9.2-6. pp.43-46.
8 Ross. Theorem 9.10. p. 51.
9 D&D. Theorem 2.5.3. p. 46.
10 D&D. Theorem 2.5.4. p. 47; Ross. Thm 10.2. p. 55.
11 Ross. Theorem 10.7. p. 58.
12 Ross. Definition 11.1. p. 63.
13 Ross. Theorem 11.2. p. 67.
14 Ross. Theorem 11.3. p. 67.
15 Ross. Corollary 11.4. p. 68.
16 Ross. Definition 11.6. p. 70.
17 Ross. Theorem 11.7. p. 71.
18 Ross. Theorem 11.8. p. 72.
19 D&D. Lemma 2.6.3. p. 51.
20 D&D. Theorem 2.6.4. p. 52.
21 Donsig, 825 Notes, 09/17.
22 D&D. Definition 2.7.2. p. 56; Ross. Definition 10.8. p. 60.
23 D&D. Definition 2.7.3. p. 56.
24 D&D. Theorem 2.7.4. p. 56; Ross. Theorem 10.11. p. 61.
25 D&D, Def 3.1.1. p. 66.
26 D&D, Example 3.1.3. p. 68.
27 Donsig, 825 Notes, 09/19.
28 D&D Theorem 3.1.5. p. 69; Ross, Definition 14.3, Theorem
14.4. pp. 92-93.
29 D&D, Prop 3.2.1., p. 71.
30 D&D, Theorem 3.2.2., p. 71.
31 H,W&T, Example 3, p. 525.
32 D&D, Theorem 3.2.3. p. 71; Ross, Theorem 14.6. p. 93.
33 Ross, Theorem 14.8. p. 94.
34 D&D, Theorem 3.2.4., p.72.; Ross, Theorem 14.9. p. 94.
35 Donsig, 825 Notes, 09/26.
36 Donsig, 825 Notes, 09/26.
37 D&D, Definition 3.2.5. p. 72.
38 Donsig, 825 Notes, 09/26. D&D. p. 77.
39 D&D. Theorem 3.2.6. p. 72.
40 D&D. Definition 3.4.2. p. 81.
41 D&D. Definition 3.4.4. p. 81.
42 D&D. Lemma 3.4.7. p. 82.
43 D&D, Theorem 3.4.8. p. 83.
44 D&D. Lemma 3.4.9. p. 85.
45 D&D. Theorem 3.4.10. p. 85.
46 D&D. Problem 3.4.G. Solution in 825 Problem Set 5.
47 D&D. p.89.
48 D&D. Theorem 4.1.1. p.89.
49 D&D. Theorem 4.1.2. p.89.
50 D&D. p.91.
51 D&D. Lemma 4.1.3. p.91.
52 D&D. Definition 4.2.1. p.92.
53 D&D. Lemma 4.2.2. p. 93.
54 D&D. Lemma 4.2.3. p.93.
55 D&D. Definition 4.2.4. p.93.
56 D&D. Theorem 4.2.5. p.94.
57 D&D. Definition 4.3.1. p. 96.
58 D&D. 4.4.N. p. 101.
59 D&D. Proposition 4.3.3. p. 97.
60 D.S. 10/15.
61 D&D. Definition 4.3.4. p.97.
62 D&D. Definition 4.3.6. p.98.
63 D&D. Proposition 4.3.5. p.97.
64 D&D. Theorem 4.3.8. p.98.
65 D&D. Proposition 4.3.9. p. 99.
2 Dongig,
10/15.
p.99.
68 D&D. Definition 4.4.1. p. 101.
69 D&D. p.102.
70 D&D. Lemma 4.4.3. p. 102.
71 D&D. Lemma 4.4.4. p. 102.
72 D&D. Lemma 4.4.5. p. 102.
73 D&D. Theorem 4.4.6. p. 103.
74 D&D. Theorem 4.4.7. p. 103.
75 D&D. p. 104
76 D&D. Def. 5.1.1. p. 108.
77 D&D. Def. 5.1.2. p. 109.
78 D&D. Def. 5.1.3. p. 111.
79 D&D. 5.5.K. p.133.
80 D&D. Prop. 5.1.4. p. 111.
81 D&D. Cor. 5.1.5. p. 111.
82 D&D. p. 112.
83 D&D. p. 114.
84 D&D. Def 5.2.3. p.115.
85 D&D. Def. 5.2.4. p.115.
86 D&D. Ex. 5.2.2. p.114.
87 D&D. p.116.
88 D&D. Ex. 5.2.9. p.118.
89 D&D. Ex. 5.2.10. p.118.
90 D&D. Def. 5.2.5. p.116.
91 D&D. p.117.
92 D&D. p.120.
93 D&D. Thm 5.3.1. p.120.
94 D&D. Thm 5.3.2. p.121.
95 D&D. Thm. 5.3.3. p.122.
96 D&D. p.122.
97 D&D. p.122.
98 D&D. p.122.
99 D&D. Thm 5.4.3. p.125.
100 D&D. Thm 5.4.4. p.125.
101 D&D. 5.4.H. p.126.
102 D&D. Def. 5.5.1. p.127.
103 D&D. Prop. 5.5.4. p.129.
104 D&D. Cor. 5.5.5. p.129.
105 d&D. Cor. 5.5.6. p.129.
106 D&D. Thm. 5.5.9. p.131.
107 D&D. Thm. 5.6.1. p.133.
108 D&D. Cor. 5.6.2. p.134.
109 D&D. p.134.
110 D&D. Cor. 5.6.3. p.134.
111 D&D. Def. 5.7.1. p.135.
112 D&D. Prop. 5.7.2. p.135.
113 D&D. Cor. 5.7.3. p.136.
114 D&D. Cor. 5.7.4. p.136.
115 D&D. Thm. 5.7.5. p.136.
116 D&D. Thm. 5.7.6. p.137.
117 Not in the book or notes.
118 D&D. Example 5.7.8. pp. 137-140.
119 D&D Def 6.1.1. p.141; Ross Def 28.1 pp.205-6
120 D&D p.141.
121 D&D. Prop 6.1.2 p.142; Ross Thm 28.2 p.207
122 D&D Lemma 6.1.3 p.142
123 D&D Cor 6.1.4 p.143
124 Ross Thm 28.3 p.208
125 D&D Thm 6.1.6 p.145; Ross Thm 28.4 p.209
126 D&D Example 6.1.7 p.145
127 D&D Exercise 6.1.E p.146
128 D&D Thm 6.2.1. p.148; Ross Thm 29.1 p.214
129 D&D Thm 6.2.2. p.148; Ross Thm 29.2 p. 214
130 D&D Thm 6.2.3. p.149; Ross Thm 29.3 p.215
131 D&D Cor 6.2.5. p.150
132 Ross Cor 29.7 p217
133 Ross Cor 29.4 p.216
134 Ross Cor 29.5 p.216
67 D&D.
Mike Janssen
NOTES
13
COUNTEREXAMPLES
Analysis Study Guide
135 D&D
149 D&D
136 D&D
150 D&D
p.151
6.2.I p.152; Ross Thm 29.8 p.217
137 Ross Thm 29.9 pp.218-9
138 D&D Def 6.3.1 p.154; Ross Def 32.1 pp.243-4
139 Ross Def 32.6 p.249
140 Ross Def 32.8 p.250
141 D&D Lma 6.3.2 p.155; Ross Lma 32.2 p.246
142 D&D Cor 6.3.3 p.155; Ross Lma 32.3 p.247
143 Ross Thm 32.4 p.247
144 D&D Def 6.3.4 p.156; Ross Def 32.1 p.244
145 D&D Thm 6.3.5 p.156; Ross Thm 32.5 p.248
146 D&D Thm 6.3.8 p.158
147 Ross Thm 32.7 p.249
148 D&D Thm 6.3.8, (4) p.158
13
Thm 6.3.9 p.160; Ross Thm 33.1 p.253
Thm 6.3.10 p.160; Ross Thm 33.2 p.254
151 D&D 6.3.S p.164
152 Ross Thm 33.3 p.254
153 Ross Thm 33.4 p.257
154 Ross Thm 33.5 p.257
155 Ross Thm 33.6 p.258
156 D&D Thm 6.4.1 p.165; Ross Thm 34.1 p.262
157 D&D p.165
158 D&D Cor 6.4.2 p.165; Ross Thm 34.3 p.264
159 D&D Lma 6.4.3 p.165
160 D&D p.167; Ross Thm 34.2 p.263
161 D&D p.167; Ross Thm 34.4 p.265
162 D&D 6.4.C p.168; Ross Thm 33.9 p.259
Counterexamples
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13
COUNTEREXAMPLES
Analysis Study Guide
Mike Janssen
13
COUNTEREXAMPLES
Analysis Study Guide
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