Ongoing Billy Flashcards
1
Q
define a metric space (X,d)
A
(X,d) consists of a non empty set X and a non-negative real valued metric d:XxX → ℝ* which satisfies the axioms.
- d(x,y)=0 x=y Ɐx,y ∈ X
- d(x,y)=d(y,x) Ɐx,y ∈ X
- d(x,z) ≤ d(x,y)+d(y,z) Ɐx,y,z ∈ X
2
Q
A metric is also known as ?
A
the distance function
3
Q
define a Subspace
A
Given any subset W⊆X, the restriction of d to W determines the subspace (W, d:= d_w) of (X,d).
4
Q
Define an open ball
A
For any metric space (X,d), the open ball of radius r>0 around point x∈X is the subspace,
B_r(x) = {y : d(y,x) < r} .
5
Q
Define a closed ball
A
For any metric space (X,d), the open ball of radius r>0 around point x∈X is the subspace,
Ɓ_r(x) = {y : d(y,x) ≤ r} .
6
Q
Define an Euclidean n-space (ℝ^n,d_2)
A
Euclidean n-space (ℝ^n,d_2) consists of all real n-dimensional vectors x=(x_1,…,x_n), equipped with the Euclidean metric
d_2(x,y) = [(x_1 - y_1)²+ … +(x_n - y_n)²]^½
where the positive square root is understood.
7
Q
what is the Euclidean metric
A
d_2(x,y) = [(x_1 - y_1)²+ … +(x_n - y_n)²]^½
8
Q
What is the open ball of the Euclidean metric?
A
B_r(x) = {y ∈ℝ^n : (⅀(y_i - x_i)^2)^½ < r} .
9
Q
What is the closed ball of the Euclidean metric?
A
Ɓ_r(x) = {y ∈ℝ^n : (⅀(y_i - x_i)^2)^½ ≤ r} .
10
Q
what is the taxicab metric?
A
d_1(x,y) = |x_1 - y_1|+ … + |x_n - y_n|
11
Q
what is the discrete metric?
A
for any non empty set X,
d(x,y) = {0 if x=y
{1 otherwise
12
Q
what is the open Ball of the discrete metric?
A
B_r(x) = { {x} if r ≤ 1
{ X if r > 1
13
Q
what is the closed Ball of the discrete metric?
A
Ɓ_r(x) = { {x} if r < 1
{ X if r ≥ 1
14
Q
define isometry
A
For any two metric spaces (X, d_X) and (Y, d_Y ), a bijection
f:X→Y is an isometry whenever d_X(x,y) = d_Y(f(x), f(y)) for all Ɐx,y ∈ X
15
Q
define isometry informally.
A
an isometry and its inverse are distance-preserving functions
16
Q
define the standard metric d_ℂ
A
d_ℂ on the complex numbers ℂ is given by
d_ℂ(z,z’) = |z-z’|
where |z|= r
17
Q
what are the balls of the standard metric, d_ℂ, like?
A
the balls in ℂ are genuine discs of radius r.
18
Q
define a Graph. Γ=(V,E)
A
Γ=(V,E) consists of a set V of vertices, and a set E of edges; each edge vw ∈E may be interpreted as joining two vertices v,w ∈V.
19
Q
define a path, in a graph Γ
A
a path in Γ from u to w is a finite sequence of edges
π(u,w) = (uv_1, v_1v_2, ….. , v_(n-2)v_(n-1), v_(n-1)w).
20
Q
whats the length of path π
A
l(π(u,w)) = n
21
Q
whats the edge metric e on the vertex set V of a path connected graph is defined by ?
A
e(u,w) = min l(π(u,w))
_π(u,w)
22
Q
what is the open ball in the edge metric
A
B_r(u) = {w∈V : π(u,w) exists with <r edges}
23
Q
what is the closed ball in the edge metric
A
Ɓ_r(u) = {w∈V : π(u,w) exists with ≤r edges}.
24
Q
in the edge metric, what is B_1(u) equal to?
A
B_1(u) = Ɓ_½(u) = {u}
25
whats the d_min on X
d_min(x,y) = {0 if x=y
| {½^n if n=min{m:x_m≠y_m}
26
Define bounded
| in terms of a function
A real-valued function f on a closed interval [a,b]⊂ℝ is bounded whenever there exists a constant K such that |f(x)|≤K for every x∈[a,b].
but the converse it false
27
whats the sup metric
d_sup(f,g) = sup |f(x)-g(x)|
| x∈[a,b]
28
what is ℬ[a,b] ?
It is the metric space (X,d_sup) for any interval [a,b]⊆ℝ.
29
what is ℒ_1[a,b] ?
The metric space (Y,d_1), where d-1 is the metric that measures the area between functions.
d_1(f,g) = ∫ |f(t)-g(t)| dt
30
what is ℒ_2[a,b] ?
The metric space (Y,d_2), where d_2 equals
d_2(f,g) = ( ∫ ( f(t)-g(t) )² dt)^½
31
what is the interval metric, d_H on X.
X is the set of all closed intervals [a,b] in the euclidean line.
d_H([a,b],[r,s]) = max{ | r - a |, | s - b | }.
32
what is the l_1 metric d_1
d_1( (a_i),(b_i) ) = ⅀|a_i - b_i|
33
in terms of d_1 what are the balls of the l_1 metric
B_r((a_i)) = {(b_i) : ⅀|a_i - b_i| < r }
and
Ɓ_r((a_i)) = {(b_i) : ⅀|a_i - b_i| ≤ r }
34
define the cartesian product of two metric spaces (X,d) and (X',d').
the cartesian product of two metric spaces (X,d) and (X',d') is the set XxX' equipped with one the metrics
1. d_a((x,x'),(y,y')) = d(x,y) + d'(x',y')
2. d_b((x,x'),(y,y')) = (d(x,y)² + d'(x',y')²)^½
3. d_c((x,x'),(y,y')) = max{d(x,y), d'(x',y')}
for any x,y∈ X and x',y'∈X'
35
define Lipschitz equivalent.
| in terms of metrics
two metrics d and e on X are lipschitz equivalent whenever there exists positive constants h,k∈ℝ such that;
he(x,y) ≤ d(x,y) ≤ ke(x,y)
for every x,y∈ X
36
Theorem 1.36. the relation between d_a, d_b, d_c?
the metrics d_a, d_b, d_c on XxX' are Lipschitz equivalent. proof?
37
♜CHAPTER.♜
♜ 2 ♜
38
Define an interior point u
An Interior point u∈U is one for which there exists ε>0 such that B_ε(u)⊆U.
39
Define the Interior of U
The Interior of U is the subset U°⊆U of all interior points.
40
define when U is open in X
when U°=U.
41
prove that every ball B_r(x) is open in X
-
42
```
theorem 2.5. given any two subsets U,V ⊆ X, the following hold;
?
?
?
?
```
1. U⊆V implies U°⊆V°
2. (U°)° = U°
3. U° is open in X
4. U° is the largest subset of U that is open in X
43
theorem 2.6.?
?
44
Define closure point.
a point x∈X is a closure point of U⊆X if B_ε(u)⋂U is non-empty for every ε>0.
45
Define closure.
The Closure of U is the superset Ū⊇U of all closure points.
46
Define closed
U is closed in X if Ū=U.
47
a set is closed in X iff ??
its complement U=X\V is open.
| proof?
48
corollary 2.9.
| every closed ball Ɓ_r(x) is ??
every closed ball Ɓ_r(x) is closed in X.
| proof?
49
Define a Partially open ball in X
a partially open ball in X is a set P_r(x) = B_r(x)⋃P, where P and Q are non-empty disjoint subsets satisfying P⋃Q={p:d(x,p)=r}.
50
```
theorem 2.13. given any two subsets U,V⊆X, the following hold:
?
?
?
?
```
1. U⊆V implies Ū ⊆ Ṽ (that squiggle should be straight)
2. V(double line) = Ṽ
3. Ṽ is closed in X
4. Ṽ is the smallest set containing V that is closed in X
51
theorem 2.14. ?
The sets X and ∅ are closed in X; so are an arbitrary intersection V = ∩V_i of closed sets V_i and a finite union V' =V'_1 ∪ V'_2 ∪.....∪V'_m of closed sets V'_j.
52
Define a sequence
For any metric space X = (X, d), a sequence in X is a function s: N → X
53
Define converges
```
A sequence (x_n) converges to the point x ∈ X whenever
∀ε > 0, ∃N ∈ N such that n ≥ N ⇒ d(x, x_n) < ε
```
54
Define convergence in terms of open balls
x_n → x whenever
| ∀ε > 0, ∃N∈N such that n ≥ N ⇒ x_n ∈ B(x).
55
Theorem 2.18. In any a metric space (X, d), the limit of a convergent sequence is ??
unique.
56
Theorem 2.19. Suppose that Y ⊆ X and y ∈ X; then y lies in Y iff ???
there exists a sequence (y_n) in Y such that y_n → y as n → ∞.
57
Define a Cauchy sequence
In any metric space (X; d), a Cauchy sequence (x_n) satisfi es
∀ε > 0, ∃N ∈N such that m,n ≥N ⇒ d(x_m, x_n) < ε
58
Proposition 2.23. If x_n → x in (X, d), then (x_n) is a ??
Cauchy sequence.
59
Define Dense
A subset Y is dense in (X, d) whenever Ȳ = X.
60
Define Bounded
| in terms of a metric space
A subset A of a metric space (X,d) is bounded whenever
| there exists x_0 ∈X and M∈R such that d(x, x_0) ≤M for every x∈A.
61
A function f : S→X is bounded whenever ?
its image f(S)⊂X is a bounded, for any set S.
62
Define the diameter
diam(A) of a bounded non-empty subset A⊆X is the real number
| sup {d(x,y) : x,y∈A}
63
Define a Boundary Point
A boundary point x∈X of A is one for which every open
| ball B_ε(x) meets both A and X\A.
64
Define a the Boundary ∂A
The boundary ∂A of A is the set of all boundary points.
65
The boundary of any subset A is ?
the set Ᾱ\A°
66
Theorem 2.31. Any subset A of (X,d) satisfi es ...
| three conditions about boundaries
1. A\∂A = Ᾱ\∂A = A°
2. ∂A = ∂(X\A)
3. ∂A is closed in X
67
Define pointwise convergence
A sequence (f_n: n≥1) of such functions converges pointwise to f on D whenever the sequence of real numbers ( f_n(x) ) converges to f(x) in ℝ for every x∈D.
68
Define Uniform convergence
A sequence (f_n: n≥1) of functions converges uniformly to f on D whenever
∀ε>0, ∃N(ε) ∈N such that n≥N(ε) ⇒ | f_n(x) - f(x) |< ε
for every x∈D.
69
f_n →f uniformly on D ⇒ ?
f_n→ f pointwise on D
70
Proposition 3.6.
Let f and f_n: D→ℝ be functions on the domain D; then
f_n→f uniformly on D iff ?
sup |f_n(x) - f(x)| exists for suciently large n, and tends to 0 as n→∞.
71
Theorem 3.10. If f_n : [a,b]→ℝ is continuous for every n∈N, and f_n→f uniformly on [a,b], then ?
f is continuous on [a,b]
72
Corollary 3.11. Suppose that f_n : [a,b]→ℝ is continuous for every n∈N and that the pointwise limit of the sequence (f_n) is discontinuous on [a,b]; then?
the convergence cannot be uniform.
73
Theorem 3.13.
If f_n : [a,b]→ℝ is integrable on [a,b] for every n∈N, and
(f_n) converges uniformly on [a,b], then ?
lim ∫f_n(t)dt = ∫ lim f_n(t)dt
| the limits are bewteen a & b. and the limit is as n→∞
74
♜ CHAPTER ♜
♜ 4 ♜
75
Define continuous at x_0 in X
Given metric spaces (X,d_x) and (Y,d_y), a function f: X→Y is continuous at x_0 in X whenever
∀ε>0; ∃δ>0 such that d_x(x,x_0) < δ ⇒ d_y (f(x), f(x_0)) < ε
76
Define a map
a continuous function.
77
define when f is continuous
f is continuous at every x_0∈X, then f is continuous.
78
define when f is continuous in terms of open balls
x∈B_δ(x_0) ⇒ f(x) ∈B_δ(f(x_0))
79
Theorem 4.5.
| The function f: X→Y is continuous at w in X i ff ?
(w_n) converges to w in X ⇒ f(w_n) converges to f(w) in Y.
80
Corollary 4.6.
| The function f: X→Y is continuous iff
(w_n) converges to w in X ⇒ f(w_n) converges to f(w) in Y. for every w∈X.
81
Define the Inverse image
For any function f: X→Y and any subset U⊆Y , the inverse image f^-1(U) is the subset {x : f(x)∈U} ⊆X.
82
Theorem 4.9.
Given metric spaces (X,d_x) and (Y,d_y ), a function f: X→Y is continuous i ff ?
(about open sets)
U open in Y ⇒ f^-1(U) open in X
83
It is extremely important to understand that the image of an open set under a continuous map need not be open.
-
84
Theorem 4.10.
Given metric spaces (X,d_x) and (Y,d_y ), a function f: X→Y is continuous i ff ?
(about closed sets)
V closed in Y ⇒ f^-1(V) closed in X
85
Theorem 4.12.
If f: X→Y and g: Y→Z are continuous functions defi ned
on metric spaces (X,d_x), (Y,d_y), and (Z,d_z), then ?
the composition g∘f: X→Z is also continuous.
86
Define Lipschitz Equivalence
| in terms of metric spaces
For any two metric spaces (X,d_x) and (Y,d_y), a bijection
f: X→Y is a Lipschitz equivalence whenever there exist positive constants h,k∈ℝ such that
hd_y (f(w), f(x)) ≤ d_x(w,x) ≤ kd_y (f(w), f(x))
for all w,x∈X.
87
Define a homeomorphism
A bijection f: X→Y is a homeomorphism whenever f and
| f^-1 are both continuous.
88
isometry ⇒ Lipschitz equivalence ⇒ homeomorphism
-
89
Proposition 4.19.
| The identity map 1_x is an isometry iff
d = e;
| it is a Lipschitz equivalence iff d and e are Lipschitz equivalent in the sense of De nition 1.34.
90
Define topologically equivalent
Two metrics d and e on a set X are topologically equivalent whenever the identity function 1_x is a homeomorphism.
91
Proposition 4.21.
| Two metrics d and e on X are topologically equivalent i ff
they give rise to precisely the same open sets.
92
Define Path connected
A metric space X is path connected if every two points
x_0, x_1 ∈X admit a continuous function σ: [0,1]→X such that σ(0) = x_0 and σ(1) = x_1.
Then σ is a path from x_0 to x_1 in X.
93
Proposition 4.24.
If f: X→Y is a homeomorphism, then X is path connected
iff ?
Y is path connected.
94
Corollary 4.25.
| As subspaces of the Euclidean plane, the interval [0,1] and the unit circle S1_ are ?
not homeomorphic.
95
♜ CHAPTER ♜
♜ 5 ♜
96
Define a Covering of A
is a collection of sets Ʋ = {U_i : i ∈ I} for which
| A ⊆ ⋃U_i
97
Define a Subcovering of Ʋ
a subcovering of U is a subcollection {U_i : i∈J} which also covers A, for some J⊆I.
98
Define a open covering
If every U_i is open, then U is an open covering of A
99
Define Compact
A subset A⊆X is compact if every open covering of A contains a finite subcovering.
100
Proposition 5.4. If A is finite, then ?
it is compact.
101
Theorem 5.5. If A is compact, then ?
it is bounded.
102
Proposition 5.6.
| Any fi nite closed interval [a,b] of the Euclidean line is compact.
-
103
Theorem 5.7.
| If f: X→Y is continuous, and A⊆X is compact, then ?
the image f(A)⊆Y is also compact.
104
Theorem 5.9.
| If A_1, ...,Ar are compact subsets of X, then ?
so is A=A_1⋃...⋃A_r.
105
Define Sequentially compact
A subspace S X is sequentially compact if any in nite
| sequence (x_n : n≥0) in S has a subsequence (x_r : r≥1) that converges to a point in S
106
Lemma 5.12.
Let (x_n) be an infi nite sequence in X, and let x∈X; if, for
any ε>0, the ball B_ε(x) contains x_n for infi nitely many values of n, then ?
(x_n) contains a subsequence that converges to x in X.
107
Lemma 5.14.
| Suppose that the infi nite sequence (x_n) contains no convergent subsequences, then ?
for every x∈X there exists an ε(x) > 0 such that B_ε(x)(x)
| contains x_n for at most fi nitely many n.
108
If a subspace A⊆X is compact, then ?
```
it is closed.
and
it is sequentially compact.
and
then it is closed and bounded.
```
109
Any closed subspace V⊆X of a compact metric space (X,d) is ?
itself compact.
110
(Heine-Borel). If a subspace A⊆ℝ^n is closed and bounded, then ?
it is compact.
111
A subspace A⊂ℝ^n is compact iff ? it
is closed and bounded.
112
```
Given any compact metric space (X,d), every continuous
function f:X→ℝ ?
```
attains its bounds.
113
Define the Cantor set K
The Cantor set K⊂ℝ is defi ned by the following inductive
procedure. Let K_0= [0,1], and defi ne K_1⊂K_0 by deleting its open middle third.
so K_1 = [0, ⅓]⋃[ ⅔, 1] is the union of two closed intervals, each of length ⅓.
Let K_n be the union of 2^n closed intervals, each of length ⅓^n, and de fine K_n+1 by deleting the open middle third of each; so K_n+1 is the union of 2^n+1 closed intervals, each of length ⅓^(n+1).
then
K= K_1 ⋂ ... ⋂ K_n ⋂...
114
The Cantor set K consists of ?
all real numbers which have
| ternary expansions containing only 0s and 2s.
115
The Cantor set is ?
uncountable
closed
compact
116
The cantor set has ?
a boundary ∂K =K in ℝ
117
♜ CHAPTER ♜
♜ 6 ♜
