MATH 425 FINAL PREP SHEET
1. Lecture 1-5
1.1. Lecture 1. EUCLIDEAN SPACE
Definition 1. Consider the set of k-tuples Rk with addition, scalar multiplication defined as usual.
This is called a vector space over the reals.
Pk
Definition 2. The inner product of two vectors x, y ∈ Rk is given as x · y = ⟨x, y⟩ = j=1 xj yj ∈ R.
1/2
P
k
2
∈ R≥0 .
Definition 3. Given x ∈ Rk , the norm of x is ∥x∥ = (x · x)1/2 =
j=1 xj
Lemma . For any x, y, z ∈ Rk , α ∈ R, we have ∥x∥ ≥ 0, ∥x∥ = 0 iff x = 0, ∥αx∥ = |α| · ∥x∥,
∥x · y∥ ≤ ∥x∥ · ∥y∥ (Cauchy-Schwartz), ∥x + y∥ ≤ ∥x∥ + ∥y∥ (Triangle).
Proposition 4. Let x ∈ Rk . x = 0 if x · z = 0 for any z ∈ Rk .
1.2. Lecture 2. SEQUENCES IN Rd
d
d
Definition 5. Let {xn }∞
n=1 ⊂ R be a sequence of points, let x ∈ R . We say xn → x as n → ∞ if
+
for any ε > 0, there exists N ∈ Z such that n ≥ N implies |xn − x| < ε. We also say that {xn }
converges to x in this situation.
Proposition 6. {xn } converges to x iff the components of xn converges to the components of x.
METRIC SPACES
Definition 7. Let X be a set, d : X × X → [0, ∞) satisfying
(a) d(p, p) = 0 for all p ∈ X
(b) d(p, q) > 0 if p ̸= q for all p, q ∈ X
(c) d(p, q) = d(q, p) for all p, q ∈ X
(d) (Triangle Inequality) d(p, q) ≤ d(p, r) + d(r, q) for all p, q, r ∈ X
Then p ∈ X are called points, d is called a metric. The pair (X, d) is called a metric space.
Proposition 8. {xn } converges to x iff {xn · z} converges to x · z for any z ∈ Rk .
1.3. Lecture 3. NEIGHBORHOODS, OPEN/CLOSED SETS
Definition 9. Let (X, d) be a metric space. A neighborhood (nbhd) of a point p ∈ X is a set Nr (p)
consisting of all q ∈ X such that d(p, q) < r for some r ∈ R+ . In set notation, Nr (p) = {q ∈ X |
d(p, q) < r}.
Definition 10. A point p ∈ X is a limit point of the set E ⊂ X if every nbhd of p contains a point
q ̸= p such that q ∈ E. In other words, for any r > 0, (Nr (p) \ {p}) ∩ E ̸= ∅.
If p ∈ E and p is not a limit point of E, it’s called an isolated point.
Definition 11. A set E is closed if every limit point of E belongs to E.
Definition 12. A point p ∈ E is an interior point of a set E ⊂ X if there exists a nbhd Nr (p) of p
such that Nr (p) ⊂ E.
Definition 13. E is open if every point of E is an interior point of E.
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2
MATH 425 FINAL PREP SHEET
1.4. Lecture 4.
Lemma . Any neighborhood Nr (p) of a point p ∈ X is open.
Definition 14. The complement of a set E, denoted E c , is the set of all points p ∈ X such that
p∈
/ E. Thus E c = {x ∈ X | x ∈
/ E}.
Theorem 15. E is open iff E c is closed. E is closed iff E c is open.
c T
S
= α∈I Eαc .
Theorem 16. Let {Eα }α∈I be a collection of sets. Then
α∈I Eα
Theorem 17.
(a) For any collection {Gα }α∈I of open sets, their union is open.
(b) For any collection {Fα }α∈I of closed sets, their intersection is closed.
(c) For any finite collection G1 , . . . , Gn of open sets, their intersection is open.
(d) For any finite collection F1 , . . . , Fn of closed sets, their union is closed.
Definition 18. Let E ⊂ X. Then E ′ is the set of limit points of E. The closure of E is then
E = E ∪ E′.
Definition 19. Let E ⊂ X. The interior of E (E ◦ ) is the set of all interior points of E.
1.5. Lecture 5.
Lemma . If 0 < s1 < s2 then Ns1 (x) ⊂ Ns2 (x).
Lemma . Nr (p) = {x ∈ Rd | ∥x∥ ≤ r}.
Theorem 20. Let (X, d) be a metric space, and E ⊂ X, then
(a) E is closed
(b) E = E iff E is closed
(c) If E ⊂ F , then E ′ ⊂ F ′
(d) If F ⊂ X is closed and E ⊂ F , then E ⊂ F
2. Lecture 6-10
2.1. Lecture 6.
Definition 21. Let (X, d) be a metric space. If E ⊂ X then the the interior of E is defined by
Int(E) = E o = {x ∈ E | x is an interior point of E}
Theorem 22. Let (X, d) be a metric space, and E ⊂ X, then
(a) E o is open
(b) E is open iff E = E o
(c) If G ⊂ E and G is open, the G ⊂ E o
(d) (E o )c = E c
Definition 23. Let E ⊂ Y ⊂ X where X is a metric space. We say that E is open relative to Y
if for every p ∈ E, ∃r > 0 such that q ∈ E wherever
d(p, q) < r,
q ∈ Y.
Definition 24. In (X, d), Y ⊂ X, p ∈ Y , Nr (p) ∩ Y in (X, d) is the same thing as Nr (p) in (Y, d).
Theorem 25. Suppose Y ⊂ X. A subset E ⊂ Y is open relative to Y iff ∃G open such that
E =G∩Y.
Definition 26. E is perfect if E is closed and every point is a limit point.
Definition
27. A collection of sets {Gα }α∈I is an open cover of a set, E, if each Gα is open and
S
E ⊂ α∈I Gα .
MATH 425 FINAL PREP SHEET
3
2.2. Lecture 7.
Definition 28. A set E is bounded if there is a real number, M , and a point, q ∈ X, such that
d(p, q) < M for all p ∈ E.
Definition 29. A set E is dense in X if E = X.
Definition 30. Let E ⊂ X. An open cover of E is a collection {Gα }α∈I of open sets in X such
that
[
E⊂
Gα .
α∈I
Definition 31. Let {Gα }α∈I be an open cover of E. A subcover is a subcollections of {Gα }α∈I ,
{Vβ }β∈J ⊂ {Gα }α∈I
such that {Vβ }β∈J is a cover of E.
Definition 32. A set K is compact if for every open cover of K, {Gα }α∈I , there exists a finite
subcollection, {Gi }ni=1 ⊂ {Gα }α∈I such that
n
[
K⊂
Gi .
i=1
Theorem 33. Suppose K ⊂ Y ⊂ X. Then K is compact relative to X iff K is compact relative to
Y.
Lemma . Let E ⊂ X be compact, then E is closed.
Lemma . Let E ⊂ K ⊂ X, K be compact. If E is closed, then E is compact.
2.3. Lecture 8.
Definition 34. Let K ⊂ X be a subset of a metric space. K is limit point compact if the
following implication holds: If E is an infinite subset of K, then E has a limit point in K.
Theorem 35. If K is compact, then K is limit point compact.
Theorem 36. Let {Kn }∞
n=1 be a sequence of nested, non-empty compact sets (Kn ⊃ Kn+1 . Then,
∞
\
Kn ̸= ∅
n=1
k
Lemma . Closed cubes in R are compact.
Theorem 37. Heine-Borel. If a set E ⊂ Rk has one of the following three properties, it also has
the other two properties:
(a) E is closed and bounded.
(b) E is compact.
(c) E is limit point compact.
2.4. Lecture 8.
Definition 38. Two subsets, A and B, of a metric space, X, are said to be separated if
A∩B =∅
and A ∩ B = ∅
Definition 39. A set is said to be connected if E is not the union of 2 nonempty separated sets.
Definition 40. A metric space (X, d) is called separable if it contains a countable, dense subset.
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MATH 425 FINAL PREP SHEET
Definition 41. A collection {Vα }α∈I of open subsets of X is said to be a base for X if the following
is true: For every x ∈ X, and every open G ⊂ X satisfying x ∈ G, there exists α ∈ I such that
x ∈ Vα ⊂ G.
Theorem
42. Let {Vα } be a base for a metric space, (X, d), then for any G ⊂ X open, G =
S
V
.
Vα ⊂G α
Theorem 43. If a metric space is separable, then it has a countable base.
2.5. Lecture 10.
Theorem 44. If a metric space (X, d), is limit point compact, then X is separable.
Theorem 45. Let (X, d) be a metric space. If X is limit point compact, then X is compact.
3. Lecture 11-15
3.1. Lecture 11. SEQUENCES IN METRIC SPACES
+
Definition 46. A sequence in a metric space (X, d) is a function P = {pn }∞
n=1 : Z → X.
Definition 47. A sequence {pn } is said to converge if there exists p ∈ X such that for all ε > 0
there exists N ∈ Z+ such that n ≥ N implies d(pn , p) < ε. This is denoted as pn → p as n → ∞ or
limn→∞ pn = p.
Theorem 48. Let {pn }∞
n=1 be a sequence in a metric space (X, d).
(a) {pn } converges to p iff every nbhd of p contains all but possibly finitely many p.
(b) If p, q ∈ X and limn→∞ pn = p and limn→∞ pn = q, then p = q.
(c) If {pn } converges then it’s bounded.
(d) If E ⊂ X and p is a limit point of E, then there is a sequence {pn } in E that converges to p.
3.2. Lecture 12. CONVERGENT SEQUENCES IN METRIC SPACES
Definition 49 (Cauchy Sequences). A sequence {pn } in a metric space (X, d) is said to be a Cauchy
sequence if for every ε > 0 there exists a N ∈ Z+ such that m, n ≥ N implies d(pn , pm ) < ε.
Proposition 50. {pn } being convergent implies that {pn } is Cauchy.
Definition 51 (Complete Metric Spaces). A metric space is ”complete” if every Cauchy sequence
is convergent.
Theorem 52. If a metric space (X, d) is compact, then it’s complete.
FUNCTIONS
Definition 53. Let (X, dx ) and (Y, dy ) be metric spaces. Let p ∈ X, E ⊂ X, q ∈ Y , f : E → Y .
Suppose p is a limit point of E. We say limx→p f (x) = q when for all ε > 0, there exists δ > 0 such
that 0 < dx (x, p) < δ implies dy (f (x), q) < ε for any x ∈ E.
Theorem 54. limx→p f (x) = q iff for all sequences {pn } ⊂ E such that pn → p we have limn→∞ f (pn ) =
q.
REAL-VALUED FUNCTIONS
Proposition 55. Consider f : E → R and g : E → R where E ⊂ X. Let p be a limit point of E.
Then
(a) limx→p (f (x) + g(x)) = limx→p f (x) + limx→p g(x)
(b) limx→p (f (x)g(x)) = (limx→p f (x)) (limx→p g(x))
lim
f (x)
(x)
(c) limx→p fg(x)
= limx→p
if limx→p g(x) ̸= 0.
x→p g(x)
MATH 425 FINAL PREP SHEET
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3.3. Lecture 13. CONTINUOUS FUNCTIONS
Definition 56 (Continuous Functions). Let E ⊂ X, f : E → Y , p ∈ E. f is continuous at p if for
all ε > 0 there exists a δ > 0 such that dx (x, p) < δ implies dy (f (x), f (p)) < ε. If f is continuous at
every point in E, then we say f is continuous on E.
Remark 57. If p ∈ E is an isolated point, f is always continuous at p. If p is a limit point of
E, then f is continuous at p iff limx→p f (x) = f (p) (or equivalently, limn→∞ f (xn ) = f (p) for all
sequences {xn } ⊂ E converging to p).
Proposition 58. Let X, Y, Z be metric spaces, let f : X → Y be continuous at p ∈ X and let
f : Y → Z be continuous at f (p) ∈ Y . Then the function h : X → Z defined by h(z) = g(f (x)) is
continuous at p ∈ X.
TOPOLOGICAL NOTIONS OF CONTINUITY
Theorem 59. f : X → Y is continuous iff f −1 (U ) is open for any U ⊂ Y open.
Theorem 60. Let fj : X → R be real-valued functions for j ∈ {1, . . . , k}. Let f (x) = (f1 (x), . . . , fk (x)) ∈
Rk such that f : X → Rk . Then f is continuous iff fj is continuous for all j.
Theorem 61. Let f : X → Y be continuous. If X is compact, then f (X) is compact.
3.4. Lecture 14.
Corollary 62. f : X → Y , f continuous on X, X compact. Then, f (X) is closed and bounded.
Corollary 63. f : X → Y , X compact, f bijective, f continuous. Then, the inverse function, f −1
is defined by
f −1 ((f (x)) = x
is continuous.
Theorem 64. Extreme Value Theorem Let f : X → R be continuous and X compact. Define
M :=⊃ {f (x) | x ∈ X} = supx∈X f (x)
m := inf{f (x) | x ∈ X} = inf f (x).
x∈X
Then there exists p, q ∈ X such that f (p) = M , f (q) = m.
Definition 65. Let f : X → Y be a function. We say that f is uniformly continuous if for all
ϵ > 0, there exists δ > 0 such that d(x, y) < δ, then d(f (x), f (y)) < ϵ.
Theorem 66. Let f : X → Y be continuous and X compact. Then, f is uniformly continuous.
3.5. Lecture 15.
Theorem 67. Let E ⊂ Rk . If E is not compact, then
(a) There exists f : E → R such that f is continuous but not bounded.
(b) There exists f : E → R such that f is continous and bounded but f doesn’t attain its
supremum.
Theorem 68. Let f : X → Y be continuous. If E ⊂ X is connected, then f (E) is connected.
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MATH 425 FINAL PREP SHEET
4. Lecture 16-20
4.1. Lecture 16. Continuous and convergent function.
Example 69. We could define fn : [0, 1] → [0, 1] by fn (x) = xn . But the pointwise limit of these
functions are not continuous again. This suggests that we need a stronger version of continuous.
Definition 70. Uniform Convergence:...
Another equivalent definition is fn converges uniformly to f if and only if
Mn := supx∈E |fn (x) − f (x)| → 0.
Theorem 71. The sequence of functions {fn } defined on E, converges uniformly on E if and only
if (it’s uniformly Cauchy).
Theorem 72. Suppose {fn }∞
n=1 is a sequence of functions defined on E, and suppose
supx∈E |f (x)| ≤ Mn
for some sequence
{Mn }∞
n .
Then
∞
X
fn (x)
n=1
converges uniformly on E if
P∞
n=1
Mn converges.
Theorem 73.
lim lim fn (t) = lim lim fn (t)
t→x n→∞
n→∞ t→x
given fn → f uniformly.
4.2. Lecture 17.
Theorem 74. If {fn } is a sequence of continuous functions on E and fn → f uniformly on E then
f is continuous. For necessity of uniformly continuous, we could consider Example 46.
CONTINUOUS EXTENSIONS
Proposition 75. Let f : X → Y , g : X → Y be continuous. Let E ⊂ X be a dense subset. Then
f (E) is dense in f (X), and if g(p) = f (p) for all p ∈ E then g(p) = f (p) for all p ∈ X.
Lemma (Urysohn’s Lemma). Let E, F ⊂ X, E, F closed, E ∩ F = ∅. Then there exists f : X →
[0, 1] such that f is continuous, f = 0 on E and f = 1 on F .
Theorem 76 (Tietze Extension Theorem). Let (X, d) be a metric space, E ⊂ X be closed. Suppose
f : E → R is continuous. There exists g : X →→ such that g is continuous and g(x) = f (x) for all
x ∈ E.
4.3. Lecture 18. no new theorems/definitions here
4.4. Lecture 19. THE SPACE OF CONTINUOUS REAL-VALUED FUNCTIONS
Definition 77. Let (X, d) be a metric space. Let C(X) = {f : X → R | f is continuous and bounded}.
Defined the function ∥ · ∥∞ : C(X) → [0, ∞) by ∥f ∥∞ = supp∈X |f (p)|.
Proposition 78. The function d : C(X) × C(X) → [0, ∞) defined by d(f, g) = ∥f − g∥∞ is a metric
on C(X).
MATH 425 FINAL PREP SHEET
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Theorem 79. (C(X), d) as defined above is a complete metric space.
VECTOR SPACES
Definition 80.
(a) A nonempty set V ⊂ Rn is a vector space if x + y ∈ V and cx ∈ V for all
x, y ∈ V and c ∈ R.
(b) If x1 , . . . , xk ∈ Rn and c1 , . . . , ck are scalars, then c1 x1 + · · · + ck xk is called a linear combination of x1 , . . . , xk . If S ⊂ Rn and if E = {all linear combinations of S} then E is the span
of S. This is denoted by E = span(S), where span(S) is also a vector space.
(c) A set containing x1 , . . . , xk ∈ Rn is said to be (linearly) independent if the relation c1 x1 +
· · · + ck xk = 0 implies c1 = · · · = ck = 0. Otherwise, {x1 , . . . , xk } is (linearly) dependent.
(d) If a vector space V contains and independent set of k vectors, but no independent set of
k + 1 vectors, V has dimension k (dim(V ) = k).
(e) An independent subset of V which also spans V is called a basis of V .
(f) The vectors e1 = (1, 0 . . . , 0), e2 = (0, 1, 0, . . . , 0), and so on form a basis {e1 , . . . , en } of Rn ,
which is referred to as the standard basis.
Theorem 81. Let k ∈ Z+ . If a vector space V is spanned by a set of k vectors, then dim(V ) ≤ k.
4.5. Lecture 20.
Corollary 82. dim(Rn ) = n.
Theorem 83. Suppose V is a vector space, and dim(V ) = n. Then,
(a) Let E ⊂ V be a set of n vectors. E spans V iff E is linearly independent.
(b) V has a basis and every basis consists of n vectors.
(c) If 1 ≤ k ≤ n and {y1 , y2 , . . . , yk } is an independent set in V . Then, V has a basis containing
{y1 , y2 , . . . , yk }.
Definition 84. A mapping A, of a vector space, X, into a vector space, Y , is said to be a linear
transformation if
A(x1 + x2 ) = Ax1 + Ax2
and
A(c · x) = c · Ax
for x, x1 , x2 ∈ X and c ∈ R.
Theorem 85. A linear operator, A, on a finite dimension space, X, is one-to-one iff the range of
A is all of X.
5. Lecture 21-24
5.1. Lecture 21.
Theorem 86. A linear operator A on a finite-dimensional vector space X is injective iff it’s surjective.
Definition 87. The range of A is denoted as R(A) = {y ∈ X | Ax = y for some x ∈ X}.
Definition 88. Let X, Y be vector spaces. We denote L(X, Y ) = {A : X → Y | A is linear} and
L(X) = L(X, X). If A1 , A2 ∈ L(X, Y ) and c1 , c2 ∈ R, c1 A1 + c2 A2 is defined by (c1 A1 + c2 A2 )(x) =
c1 A1 x + c2 A2 x.
Definition 89. If X, Y, Z are vector spaces and A ∈ L(X, Y ), B ∈ L(Y, Z) then the product of A
and B, denoted BA ∈ L(X, Z), is the composition of A and B. In other words (BA)(x) = B(A(x)).
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MATH 425 FINAL PREP SHEET
Definition 90. For A ∈ L(Rn , Rm ), define the norm of A, denoted ∥A∥, by ∥A∥ = sup{∥Ax∥ |
∥x∥ ≤ 1, x ∈ Rn }.
Lemma . ∥Ax∥ ≤ ∥A∥ · ∥x∥. Also, ∥A∥ = inf{c > 0 | ∥Ax∥ ≤ C∥x∥ for all x ∈ Rn }.
Theorem 91.
(a) If A ∈ L(Rn , Rm ) then ∥A∥ < ∞ and A is uniformly continuous.
(b) If A, B ∈ L(Rn , Rm ), c ∈ R let ∥A + B∥ ≤ ∥A∥ + ∥B∥ and ∥cA∥ = |c| · ∥A∥ with d(A, B) =
∥A − B∥. Then (L(Rn , Rm ), d) is a metric space.
(c) If A ∈ L(Rn , Rm ) and B ∈ L(Rm , Rk ), then ∥BA∥ ≤ ∥B∥ · ∥A∥.
5.2. Lecture 22.
Theorem 92. Let ω be the set of all invertible operators on Rn .
(a) If A ∈ ω, B ∈ L(Rn ) and ∥B − A| < ∥A1−1 ∥ , then B ∈ ω.
(b) ω is an open subset of L(Rn ) and the A−1 is continuous on ω.
5.3. Lecture 23. DIFFERENTIATION
Definition 93. Let E ⊂ Rn , E open. Let f : E → Rm be a function. Let x ∈ E. If there exists a
(x)−Ah}
linear transformation A : Rn → Rm such that limh→0 ∥f (x+h)−f
= 0 then we say that f is
∥h∥
′
differentiable at x and we write f (x) = A.
Remark 94. We sometimes write f (h) = o(∥h∥) iff
f (h)
∥h∥
→ 0 as ∥h∥ → 0.
Theorem 95 (Uniqueness of Derivatives). Let f : E → Rm , E ⊂ Rn open, x ∈ E. Suppose
f ′ (x) = A1 and f ′ (x) = A2 , then A1 = A2 .
Theorem 96 (Chain Rule). Let f : E → Rm , E ⊂ Rn open, x ∈ E. Suppose f is differentiable at
x ∈ E. Let g : U → Rk , were U ⊂ Rm open and f (E) ⊂ U . Suppose g is differentiable at f (x).
Then g ◦ f : E → Rk is differentiable at x with (g ◦ f )′ (x) = g ′ (f (x)) · f ′ (x).
5.4. Lecture 24. PARTIAL/DIRECTIONAL DERIVATIVES
Definition 97. Let f : E → Rn , E ⊂ Rn open. The components of f are fj : E → R for
j = 1, . . . , m, where f (x) = (f1 (x), . . . , fm (x)).
Definition 98 (Partial Derivative). Take f and E as the definition above. Let {e1 , . . . , en } be the
f (x+tek )−fj (x)
standard basis on Rn . Then Dk fj (x) = limt→0 j
(if the limit exists).
t
Theorem 99. Let f and E be defined as above. Suppose f is differentiable at x ∈ E. Then Dk fj (x)
exists and f ′ (x) · ek = (Dk f1 (x), . . . , Dk fm (x)).
Remark 100. Take γ : (a, b) → E ⊂ Rn be a function defining a path on E. Let E,f be defined as
above, and let g(t) = f (γ(t)). Then by chain rule, g ′ (t) = f ′ (γ(t)) · γ ′ (t).
Definition 101 (Directional Derivative). Let f : E → Rn be differentiable at x ∈ E ⊂ Rn (E open).
(x)
Let u ∈ Rn be a unit vector. Then limt→0 f (x+tu)−f
= Du f (x) is the directional derivative of f
t
at x in the direction u.
Remark 102. If f ′ (x) exists, then the chain rule implies Du f (x) = f ′ (x) · u.
Definition 103 (Continuous Differentiability). A differentiable mapping f : E → Rm , E ⊂ Rn
open, is said to be continuously differentiable in E if f ′ : E → L(Rn , Rm ) is continuous. The space
of such functions is given as C 1 (E, Rm ).
Theorem 104. Suppose f : E → Rm , E ⊂ Rm open. Then f ∈ C 1 (E, Rm ) iff the partial derivatives, Dk fj exist and are continuous on E for all k, j.