Metric Spaces, Lecture Notes - Mathematics, Study notes for Calculus. University of California (CA) - UCLA

Calculus

Description: Metric spaces, Metrics a Nation of Distance, Examples, Convergence
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Metric Spaces
Adrian Down 16779577
August 2, 2005
1 Metrics: a notion of distance
So far in analysis on R, every definition (convergence, Cauchy, continuity)
has included statements like “|xy|< ”. The idea is that this |xy|
represents a distance from xto y.
A distance should satisfy the properties:
i) |xy| ≥ 0,x, y R
ii) |xy|= 0 x=y
iii) |xy|=|yx|
iv) “Triangle inequality”: |xy|+|yz| ≥ |xz|,x, y, z
Definition: a metric space is a set of “objects” Stogether with a function
don the set of pairs (x, y), x, y Ssuch that
d(x, y)R,x, y S
1)d(x, y)0,x, y S
2)d(x, y) = 0 x=y
3)d(x, y) = d(y, x),x, y S
4)d(x, y) + d(y, z)d(x, z),x, y, z S
dis called the metric on S. May write (S, d) to specify the metric.
4) is called the triangle inequality; it is the most important, and it is
usually the hardest to verify.
2 Examples
1) (R, d) is a metric space where d(x, y) = |xy|
1
2) “Discrete metric space”: Sis any set, and define
d(x, y) = (1x, y S, x 6=y
0xS, x =y
3) “Euclidean k-space”:
S={(x1, x2, . . . , xk)|xiR,1ik}
d(x,y) = k
X
i=1
(xiyi)2!
1
2
Properties 1, 2, and 3 are clear. Property 4 takes work. Best way is to
consider dot products of vectors.
This is the most important metric space that we will cover. The given
metric space is called the standard metric on Rk.
4) “Manhattan metric”:
S=Rk
d(x,y) =
k
X
i=1 |xiyi|
Properties 1, 2, and 3 are easy. For 4, use the triangle inequality on R,
X|xiyi|+X|yizi| ≥ X|xizi|
5) C[0,1] :
S={f|fis a continous function on [0,1]}
d(f, g) = sup{|f(x)g(x)||x[0,1]}
Note that f, g continuous on [0,1] fgis continuous on [0,1] d(f, g)
exists.
To verify property 2,
d(f, g) = 0 sup{|f(x)g(x)|} = 0 → |f(x)g(x)|= 0,xf(x) = g(x),x
2
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