Metric Spaces
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17 terms
Terms | 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) } |
| p-adic Metric | Let X=Z, p be prime. Define the p-adic metric: 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. |
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