metric spaces

Question 1

axioms of a metric space:

Answer 1

d(x,y)=0 <=> x=y for all x,y in X
d(x,y)=d(y,x) for all x,y, in X
d(x,z)<=d(x,y)+d(y,z) for all x,y,z in X

Question 2

open ball definition:

Answer 2

Br(x)={y:d(x,y)<r}

Question 3

closed ball definition:

Answer 3

B̄r(x)={y:d(x,y)<=r}

Question 4

euclidean n-space:

Answer 4

in the real numbers
d2(x,y)=((x1-y1)^2+…+(xn-yn)^2)^1/2
n dimensional space basically

Question 5

taxicab metric:

Answer 5

in the real numbers
d1(x,y)=|x1-y1|+…+|xn-yn|

Question 6

d∞ metric:

Answer 6

in the real numbers
d∞(x,y)=max{|x1-y1|,…,|xn-yn|}

Question 7

discrete metric:

Answer 7

on any nonempty set X
d(x,y)= 0 if x=y, 1 otherwise

Question 8

isometry of a metric space:

Answer 8

a bijection where dX(x,y)=dY(f(x),f(y)) for all x,y in X

Question 9

standard metric:

Answer 9

on the complex numbers
dC(z,z’)=|z-z’|

Question 10

when are 2 metrics lipschitz equivalent:

Answer 10

when he(x,y)<=d(x,y)<=ke(x,y), h and k being positive constants, d and e being metrics on X, for all x,y in X

Question 11

edge metric:

Answer 11

e(u,w)=min l(π(u,w)) (meaning miniumum length between vertices on a graph, smallest path)

Question 12

sup metric:

Answer 12

dsup(f,g)=sup|f(x)-g(x)| x in the domain of each, for any f,g in X

Question 13

L1 metric:

Answer 13

on [a,b]
d1(f,g)=integral b,a of |f(t)-g(t)| dt

Question 14

L2 metric:

Answer 14

on [a,b]
d2(f,g)=(integral b,a of (f(t)-g(t))^2 dt)^1/2

Question 15

interval metric:

Answer 15

on the set of all closed intervals in the euclidean line
dH([a,b],[r,s])=max{|r-a|,|s-b|}

Question 16

interior point:

Answer 16

a point u in X is an interior point of the subset U if there exists ε>0 such that Bε(u) is in U - it’s an interior point if you can draw a small ball around it that’s still within/on the edge of the area, interior points cannot be on the edge

Question 17

open set:

Answer 17

a set U is open in X if for all u in U, there exists ε>0 such that Bε(u) in U - the set of interior points = the entire set U

Question 18

fun facts about open sets:

Answer 18

all open balls are open in X
any union of 2 open balls are open in X
the complement of a closed ball is open in X
if (X,d) is discrete, any subset U is open in X

Question 19

closure points:

Answer 19

a point x in X is a closure point of a subset U in X if Bε(x) intersection U is nonempty for every ε>0 - basically same as interior points but includes the border, cause a small ball on a border point will intersect with the area clearly

Question 20

closed set:

Answer 20

a set U in X is closed if Bε(x) intersection U is nonempty for every ε>0 - if the set of closure points of U = U

Question 21

fun facts about closed sets:

Answer 21

a set is closed if its complement is open
all closed balls are closed
an intersection of closed balls is closed
the complement of an open ball is closed
if (X,d) is discrete, any set is closed

Question 22

convergence:

Answer 22

for all ε>0, there exists a natural number N such that n>=N -> d(x,xn)<ε, x is the limit of xn here

Question 23

convergence in terms of open balls:

Answer 23

for all ε>0, there exists a natural number N such that n>=N -> xn in Bε(x)

Question 24

convergence in euclidean m-space:

Answer 24

xn->x iff |xn-x|->o

Question 25

convergence in discrete metric space and graph spaces with an edge metric:

Answer 25

xn->x iff xn=x for all n>=some N

Question 26

cauchy sequence definition:

Answer 26

for all ε>0, there exists a natural number N such that m,n>=N -> d(xm,xn)<ε>x in (X,d), xn is a cauchy sequence
basically the distance between terms of the sequence gets closer and closer but no specified limit</ε>

Question 27

bounded subset:

Answer 27

a subset A is bounded when there exists x0 in X and a real number M such that d(x,x0)<=M for every x in A - when A is contained within B̄M(x0)

Question 28

diameter:

Answer 28

of a bounded nonempty set A
sup{d(x,y): x,y in A)
sometimes we say the diameter of the empty set is 0

Question 29

boundary point:

Answer 29

a point within the closure but not the interior of a set
so just the well. yknow. boundary
every open ball intersects A and X\A

Question 30

pointwise convergence:

Answer 30

for all x in D (domain) and for all ε>0 there exists natural number N such that |fn(x)-f(x)|<ε>=N
weaker than pointwise cause N can be different for different x's</ε>

Question 31

uniform convergence:

Answer 31

for all ε>0, there exists natural number N such that |fn(x)-f(x)| for all n>=N and x in D (domain)
stronger than pointwise cause the Same N works for All x

Question 32

what carries over in uniform convergence:

Answer 32

integral limits and continuity

Question 33

continuity at a point and overall:

Answer 33

(for all x0 if for whole function), for all ε>0, there exists δ>0 such that dX(x,x0)<δ -> dY(f(x),f(x0))<ε
alternatively, x in Bδ(x0) -> f(x) in Bε(f(x0))

Question 34

continuity and convergence:

Answer 34

a function f:X->Y is continuous iff (wn) converges to w in X -> f(wn) converges to f(w) in Y

Question 35

inverse image:

Answer 35

notation - f^-1(U)
={x:f(x) in U} contained in X

Question 36

inverse image and continuity:

Answer 36

a function between 2 metric spaces (X,dX) and (Y,dY) is continuous iff U open in Y -> f^-1(U) open in X
same for closed

Question 37

when is a bijection between 2 metric spaces a lipschitz map/equivalence:

Answer 37

when there exists positive real constants h and k such that hdY(f(w),f(x))<=dX(w,x)<=kdY(f(w),f(x)) for all w,x in X
less strict than isometry

Question 38

when is a bijection between 2 metric spaces a homeomorphism:

Answer 38

when f and f^-1 are both continuous
less strict lipschitz map so even less strict isometry

Question 39

completeness:

Answer 39

a metric space is complete if every cauchy sequence converges to a limit in X

Question 40

complete examples:

Answer 40

the real numbers are complete
open balls in euclidean n-space are not
vertex sets are complete
euclidean n-space itself is complete
a closed subspace of a complete metric space is complete
a complete subspace in X is closed in X

Question 41

contraction:

Answer 41

given a metric space (X,d), a self map f:X->X is a contraction when there exists a constant 0<K<1 such that d(f(x),f(y))<=Kd(x,y) for all x,y in X
these are all continuous

Question 42

fixed point:

Answer 42

a point on a self map f:X->X where x=f(x)
let X be complete, and f:X->X a contraction, then f has a unique fixed point