Page 52 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 1 2
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(b) d (f, g) = ë D ò é f(x) g(x) dx ù û
2

Although this does not have such case straight forward geometric interpretation as
the last example, this case turns out to be the most important in practice. It corresponds
to who doing a “least squares approximation”.

(c) d(f, g) = max {|f(x) – g(x))|| 0  x  1}
Geometrically, this is the largest distance between the graphs.

Remarks:
1. The triangle inequality does hold for these metrics
2. As in the R case one may define d for any p  1 and get a metric.
2
p
3.1.1 Space Properties

Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly
normal). An important consequence is that every metric space admits partitions of unity and
that every continuous real-valued function defined on a closed subset of a metric space can be
extended to a continuous map on the whole space (Tietze extension theorem). It is also true that
every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended
to a Lipschitz-continuous map on the whole space.
Metric spaces are first countable since one can use balls with rational radius as a neighborhood
base.
The metric topology on a metric space M is the coarsest topology on M relative to which the
metric d is a continuous map from the product of M with itself to the non-negative real numbers.

3.1.2 Distance between Points and Sets; Hausdorff Distance and
Gromov Metric

A simple way to construct a function separating a point from a closed set (as required for a
completely regular space) is to consider the distance between the point and the set. If (M, d) is a
metric space, S is a subset of M and x is a point of M, we define the distance from x to S as
d(x, S) = inf {d(x, s) : s  S}

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