Geometry and Topology Flashcards
Properties of a metric
A metric is a map d: X
x X -> R
satisfying:
* d
is positive definite: for all x,y
in X
, d(x,y) >= 0
, and d(x,y) = 0
if and only if x = y
* d
is symmetric: for all x,y
in X
, d(x,y) = d(y,x)
* d
satisfies the triangle inequality: for all x,y,z
in X
, d(x,z) <= d(x,y) + d(y,z)
Properties of a norm
A normed vector space over a field F
, is a vector field V
equipped with a map, the norm, ||-||: V -> F
satisfying:
* for all x
in V
, ||x|| >= 0
, and ||x|| = 0
if and only if x = 0
is in V
* for all a
in F
, x
in V
, ||a x|| = |a| ||x||
* for all x,y
in V
, ||x + y|| <= ||x|| + ||y||
(triangle inequality)
Lift metric
Open ball
Closed ball
Sphere
radius r
around a
Proof of a continuous map f
over metric spaces X,Y
Proof of a Euclidean isometry f
A Euclidean isometry is a map f: E^n -> E^n
which preserves the Euclidean distance
Euclidean norm
Euclidean metric / distance
Orthogonal group
Square matrices, where its transpose equals its inverse
d(1,n,inf) for vectors x,y
d(1,n,inf) for functions f,g
Lipschitz map properties
A map f: X -> Y
satisfies the Lipschitz condition if:
Contraction (mapping theorem)
A contraction is a Lipschitz function with constant < 1
:
Proof of a sequence {x(n)}
converging to x
Completeness of a metric space (X,d)
The metric space is complete if every Cauchy sequence in X
converges to a point in X
Cauchy sequence definition
Diameter of a (sub)set A