## Metric Spaces

##### Created by:

as2187  on July 15, 2012

Pop out
No Messages

# Metric Spaces

 Distance FunctionA function d: X x X → R such that: (i) d(P,Q)≥0 with equality iff P=Q (ii) d(P,Q)=d(Q,P) ∀P,Q∈X (iii) d(P,Q) + d(Q,R) ≥ d(P,R) ∀P,Q,R∈X
1/17
Preview our new flashcards mode!

Order by

#### Definitions

Distance Function A function d: X x X → R such that:
(i) d(P,Q)≥0 with equality iff P=Q
(ii) d(P,Q)=d(Q,P) ∀P,Q∈X
(iii) d(P,Q) + d(Q,R) ≥ d(P,R) ∀P,Q,R∈X
Metric Space A metric space (X,d) consists of a set X and a distance function d.
Triangle Inequality d(P,Q) + d(Q,R) ≥ d(P,R) ∀P,Q,R∈X
Cauchy-Schwarz Inequality Let u and v be vectors in an inner product space V.
Then (u,v)²≤ ||u||²||v||² with equality holding iff u and v are scalar multiples of each other.
Discrete Metric For any set X, d(x,y) = 1 if x≠y
0 if x=y
British Rail Metric Consider Rⁿ with the Euclidean metric d. Define p on Rⁿ by:
p(P,Q) = d(P,O) + d(O,Q) if P≠Q
0 if P=Q
ie all journeys from P to Q must go via O.
Ultra-metric A metric space (X,d) is called ultra-metric if d satisfies:
d(P,R) ≤ max { d(P,Q) , d(Q,R) }
d(a,b) = 0 if a=b
p⁻ⁿ if a≠b, where n=max{ s∈N : p⁵|(a-b) }
Lipschitz equivalence Two metrics p₁and p₂on a set X are Lipschitz equaivalent if ∃ 0<λ₁≤λ₂∈R such that:
λ₁p₁≤ p₂≤ λ₂p₁
Open Ball Let (X,d) be a metric space.
Bd(P,δ) = { Q∈X : d(P,Q)<δ }
Open Set A subset U⊂X of a metric space (X,d) is called open if ∀P∈U, ∃ B(P,δ) ,with δ sufficiently small, contained in U.
Closed Set A subset F⊂X is called closed if X \ F is open.
Open Neighbourhood If P is a point in (X,d), an open neighbourhood N at P is an open subset N⊃P.
Converges Suppose x₁,x₂, ... is a sequence of points in (x,d). We say x(n) converges to x if d(x(n),x)→0 as n→∞.
Continuous f : ( X,p₁) → ( Y,p₂) is continous at x∈X if given ε>0, ∃δ>0 such that p₁(x',x) < δ implies p₂( f(x') , f(x) ) < ε.
Uniform Continuous f : ( X,p₁) → ( Y,p₂) is uniform continous on X if given ε>0, ∃δ>0 such that p₁(x₁,x₂) < δ implies p₂( f(x₁) , f(x₂) ) < ε.
Complete A metric space (X,p) is complete if for any sequence x₁,x₂, ... ∈X with the property "given ε>0, ∃N such that ∀m,n≥N , p( x(m), x(n) ) <ε", we have x(n)→x for some limit point x∈X.

### First Time Here?

Welcome to Quizlet, a fun, free place to study. Try these flashcards, find others to study, or make your own.

### Set Champions

##### Scatter Champion

32.1 secs by as2187