Ongoing Billy Flashcards
define a metric space (X,d)
(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
A metric is also known as ?
the distance function
define a Subspace
Given any subset W⊆X, the restriction of d to W determines the subspace (W, d:= d_w) of (X,d).
Define an open ball
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} .
Define a closed ball
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} .
Define an Euclidean n-space (ℝ^n,d_2)
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.
what is the Euclidean metric
d_2(x,y) = [(x_1 - y_1)²+ … +(x_n - y_n)²]^½
What is the open ball of the Euclidean metric?
B_r(x) = {y ∈ℝ^n : (⅀(y_i - x_i)^2)^½ < r} .
What is the closed ball of the Euclidean metric?
Ɓ_r(x) = {y ∈ℝ^n : (⅀(y_i - x_i)^2)^½ ≤ r} .
what is the taxicab metric?
d_1(x,y) = |x_1 - y_1|+ … + |x_n - y_n|
what is the discrete metric?
for any non empty set X,
d(x,y) = {0 if x=y
{1 otherwise
what is the open Ball of the discrete metric?
B_r(x) = { {x} if r ≤ 1
{ X if r > 1
what is the closed Ball of the discrete metric?
Ɓ_r(x) = { {x} if r < 1
{ X if r ≥ 1
define isometry
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
define isometry informally.
an isometry and its inverse are distance-preserving functions
define the standard metric d_ℂ
d_ℂ on the complex numbers ℂ is given by
d_ℂ(z,z’) = |z-z’|
where |z|= r
what are the balls of the standard metric, d_ℂ, like?
the balls in ℂ are genuine discs of radius r.
define a Graph. Γ=(V,E)
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.
define a path, in a graph Γ
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).
whats the length of path π
l(π(u,w)) = n
whats the edge metric e on the vertex set V of a path connected graph is defined by ?
e(u,w) = min l(π(u,w))
_π(u,w)
what is the open ball in the edge metric
B_r(u) = {w∈V : π(u,w) exists with <r edges}
what is the closed ball in the edge metric
Ɓ_r(u) = {w∈V : π(u,w) exists with ≤r edges}.
in the edge metric, what is B_1(u) equal to?
B_1(u) = Ɓ_½(u) = {u}
whats the d_min on X
d_min(x,y) = {0 if x=y
{½^n if n=min{m:x_m≠y_m}
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
whats the sup metric
d_sup(f,g) = sup |f(x)-g(x)|
x∈[a,b]
what is ℬ[a,b] ?
It is the metric space (X,d_sup) for any interval [a,b]⊆ℝ.
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
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
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 | }.
what is the l_1 metric d_1
d_1( (a_i),(b_i) ) = ⅀|a_i - b_i|
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 }
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
- d_a((x,x’),(y,y’)) = d(x,y) + d’(x’,y’)
- d_b((x,x’),(y,y’)) = (d(x,y)² + d’(x’,y’)²)^½
- d_c((x,x’),(y,y’)) = max{d(x,y), d’(x’,y’)}
for any x,y∈ X and x’,y’∈X’
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
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?
♜CHAPTER.♜
♜ 2 ♜
Define an interior point u
An Interior point u∈U is one for which there exists ε>0 such that B_ε(u)⊆U.
Define the Interior of U
The Interior of U is the subset U°⊆U of all interior points.
define when U is open in X
when U°=U.
prove that every ball B_r(x) is open in X
-
theorem 2.5. given any two subsets U,V ⊆ X, the following hold; ? ? ? ?
- U⊆V implies U°⊆V°
- (U°)° = U°
- U° is open in X
- U° is the largest subset of U that is open in X
theorem 2.6.?
?
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.
Define closure.
The Closure of U is the superset Ū⊇U of all closure points.
Define closed
U is closed in X if Ū=U.
a set is closed in X iff ??
its complement U=X\V is open.
proof?
corollary 2.9.
every closed ball Ɓ_r(x) is ??
every closed ball Ɓ_r(x) is closed in X.
proof?
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}.
theorem 2.13. given any two subsets U,V⊆X, the following hold: ? ? ? ?
- U⊆V implies Ū ⊆ Ṽ (that squiggle should be straight)
- V(double line) = Ṽ
- Ṽ is closed in X
- Ṽ is the smallest set containing V that is closed in X
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.
Define a sequence
For any metric space X = (X, d), a sequence in X is a function s: N → X
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
Define convergence in terms of open balls
x_n → x whenever
∀ε > 0, ∃N∈N such that n ≥ N ⇒ x_n ∈ B(x).
Theorem 2.18. In any a metric space (X, d), the limit of a convergent sequence is ??
unique.
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 → ∞.
Define a Cauchy sequence
In any metric space (X; d), a Cauchy sequence (x_n) satisfies
∀ε > 0, ∃N ∈N such that m,n ≥N ⇒ d(x_m, x_n) < ε
Proposition 2.23. If x_n → x in (X, d), then (x_n) is a ??
Cauchy sequence.
Define Dense
A subset Y is dense in (X, d) whenever Ȳ = X.
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.
A function f : S→X is bounded whenever ?
its image f(S)⊂X is a bounded, for any set S.
Define the diameter
diam(A) of a bounded non-empty subset A⊆X is the real number
sup {d(x,y) : x,y∈A}
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.
Define a the Boundary ∂A
The boundary ∂A of A is the set of all boundary points.
The boundary of any subset A is ?
the set Ᾱ\A°
Theorem 2.31. Any subset A of (X,d) satisfies …
three conditions about boundaries
- A\∂A = Ᾱ\∂A = A°
- ∂A = ∂(X\A)
- ∂A is closed in X
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.
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.
f_n →f uniformly on D ⇒ ?
f_n→ f pointwise on D
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→∞.
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]
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.
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→∞
♜ CHAPTER ♜
♜ 4 ♜
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
Define a map
a continuous function.
define when f is continuous
f is continuous at every x_0∈X, then f is continuous.
define when f is continuous in terms of open balls
x∈B_δ(x_0) ⇒ f(x) ∈B_δ(f(x_0))
Theorem 4.5.
The function f: X→Y is continuous at w in X iff ?
(w_n) converges to w in X ⇒ f(w_n) converges to f(w) in Y.
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.
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.
Theorem 4.9.
Given metric spaces (X,d_x) and (Y,d_y ), a function f: X→Y is continuous iff ?
(about open sets)
U open in Y ⇒ f^-1(U) open in X
It is extremely important to understand that the image of an open set under a continuous map need not be open.
-
Theorem 4.10.
Given metric spaces (X,d_x) and (Y,d_y ), a function f: X→Y is continuous iff ?
(about closed sets)
V closed in Y ⇒ f^-1(V) closed in X
Theorem 4.12.
If f: X→Y and g: Y→Z are continuous functions defined
on metric spaces (X,d_x), (Y,d_y), and (Z,d_z), then ?
the composition g∘f: X→Z is also continuous.
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.
Define a homeomorphism
A bijection f: X→Y is a homeomorphism whenever f and
f^-1 are both continuous.
isometry ⇒ Lipschitz equivalence ⇒ homeomorphism
-
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 Denition 1.34.
Define topologically equivalent
Two metrics d and e on a set X are topologically equivalent whenever the identity function 1_x is a homeomorphism.
Proposition 4.21.
Two metrics d and e on X are topologically equivalent iff
they give rise to precisely the same open sets.
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.
Proposition 4.24.
If f: X→Y is a homeomorphism, then X is path connected
iff ?
Y is path connected.
Corollary 4.25.
As subspaces of the Euclidean plane, the interval [0,1] and the unit circle S1_ are ?
not homeomorphic.
♜ CHAPTER ♜
♜ 5 ♜
Define a Covering of A
is a collection of sets Ʋ = {U_i : i ∈ I} for which
A ⊆ ⋃U_i
Define a Subcovering of Ʋ
a subcovering of U is a subcollection {U_i : i∈J} which also covers A, for some J⊆I.
Define a open covering
If every U_i is open, then U is an open covering of A
Define Compact
A subset A⊆X is compact if every open covering of A contains a finite subcovering.
Proposition 5.4. If A is finite, then ?
it is compact.
Theorem 5.5. If A is compact, then ?
it is bounded.
Proposition 5.6.
Any finite closed interval [a,b] of the Euclidean line is compact.
-
Theorem 5.7.
If f: X→Y is continuous, and A⊆X is compact, then ?
the image f(A)⊆Y is also compact.
Theorem 5.9.
If A_1, …,Ar are compact subsets of X, then ?
so is A=A_1⋃…⋃A_r.
Define Sequentially compact
A subspace S X is sequentially compact if any innite
sequence (x_n : n≥0) in S has a subsequence (x_r : r≥1) that converges to a point in S
Lemma 5.12.
Let (x_n) be an infinite sequence in X, and let x∈X; if, for
any ε>0, the ball B_ε(x) contains x_n for infinitely many values of n, then ?
(x_n) contains a subsequence that converges to x in X.
Lemma 5.14.
Suppose that the infinite 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 finitely many n.
If a subspace A⊆X is compact, then ?
it is closed. and it is sequentially compact. and then it is closed and bounded.
Any closed subspace V⊆X of a compact metric space (X,d) is ?
itself compact.
(Heine-Borel). If a subspace A⊆ℝ^n is closed and bounded, then ?
it is compact.
A subspace A⊂ℝ^n is compact iff ? it
is closed and bounded.
Given any compact metric space (X,d), every continuous function f:X→ℝ ?
attains its bounds.
Define the Cantor set K
The Cantor set K⊂ℝ is defined by the following inductive
procedure. Let K_0= [0,1], and define 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 define 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 ⋂…
The Cantor set K consists of ?
all real numbers which have
ternary expansions containing only 0s and 2s.
The Cantor set is ?
uncountable
closed
compact
The cantor set has ?
a boundary ∂K =K in ℝ
♜ CHAPTER ♜
♜ 6 ♜
