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.
