Measure, Integration & Real Analysis, 2021a

Section 6A Metric Spaces 147
6A
Metric Spaces
Open Sets, Closed Sets, and Continuity
Much of analysis takes place in the context of a metric space, which is a set with a
notion of distance that satisfies certain properties. The properties we would like a
distance function to have are captured in the next definition, where you should think
of d( f , g) as measuring the distance between f and g.
Specifically, we would like the distance between two elements of our metric space
to be a nonnegative number that is 0 if and only if the two elements are the same. We
would like the distance between two elements not to depend on the order in which
we list them. Finally, we would like a triangle inequality (the last bullet point below),
which states that the distance between two elements is less than or equal to the sum
of the distances obtained when we insert an intermediate element.
Now we are ready for the formal definition.
6.1 Definition metric space
A metric on a nonempty set V is a function d : V × V → [0, ∞) such that
• d( f , f )=0 for all f ∈ V;
• if f , g ∈ V and d( f , g) =0, then f = g;
• d( f , g) =d(g, f ) for all f , g ∈ V;
• d( f , h) ≤ d( f , g)+d(g, h) for all f , g, h ∈ V.
A metric space is a pair (V, d), where V is a nonempty set and d is a metric on V.
6.2 Example metric spaces
• Suppose V is a nonempty set. Define d on V × V by setting d( f , g) to be 1 if
f ̸= g and to be 0 if f = g. Then d is a metric on V.
• Define d on R × R by d(x, y) =|x − y|. Then d is a metric on R.
• For n ∈ Z + , define d on R n × R n by
d ( (x 1 ,...,x n ), (y 1 ,...,y n ) ) = max{|x 1 − y 1 |,...,|x n − y n |}.
Then d is a metric on R n .
• Define d on C([0, 1]) × C([0, 1]) by d( f , g) =sup{| f (t) − g(t)| : t ∈ [0, 1]};
here C([0, 1]) is the set of continuous real-valued functions on [0, 1]. Then d is
a metric on C([0, 1]).
• Define d on l 1 × l 1 by d ( (a 1 , a 2 ,...), (b 1 , b 2 ,...) ) = ∑ ∞ k=1 |a k − b k |; here l 1
is the set of sequences (a 1 , a 2 ,...) of real numbers such that ∑ ∞ k=1 |a k| < ∞.
Then d is a metric on l 1 .
Measure, Integration & Real Analysis, by Sheldon Axler