5Metric spaces
IB Analysis II
5.1 Preliminary definitions
Definition (Metric space). Let
X
be any set. A metric on
X
is a function
d : X × X → R that satisfies
– d(x, y) ≥ 0 with equality iff x = y (non-negativity)
– d(x, y) = d(y, x) (symmetry)
– d(x, y) ≤ d(x, z) + d(z, y) (triangle inequality)
The pair (X, d) is called a metric space.
We have seen that we can define convergence in terms of a metric. Hence,
we can also define open subsets, closed subsets, compact spaces, continuous
functions etc. for metric spaces, in a manner consistent with what we had for
normed spaces. Moreover, we will show that many of our theorems for normed
spaces are also valid in metric spaces.
Example.
(i) R
n
with the Euclidean metric is a metric space, where the metric is defined
by
d(x, y) = ∥x − y∥ =
q
X
(x
j
− y
j
)
2
.
(ii)
More generally, if (
V, ∥ · ∥
) is a normed space, then
d
(
x, y
) =
∥x − y∥
defines a metric on V .
(iii) Discrete metric: let X be any set, and define
d(x, y) =
(
0 x = y
1 x = y
.
(iv) Given a metric space (X, d), we define
g(x, y) = min{1, d(x, y)}.
Then this is a metric on X. Similarly, if we define
h(x, y) =
d(x, y)
1 + d(x, y)
is also a metric on X. In both cases, we obtain a bounded metric.
The axioms are easily shown to be satisfied, apart from the triangle
inequality. So let’s check the triangle inequality for
h
. We’ll use a general
fact that for numbers a, c ≥ 0, b, d > 0 we have
a
b
≤
c
d
⇔
a
a + b
≤
c
c + d
.
Based on this fact, we can start with
d(x, y) ≤ d(x, z) + d(z, y).
Then we obtain
d(x, y)
1 + d(x, y)
≤
d(x, z) + d(z, y)
1 + d(x, z) + d(z, y)
=
d(x, z)
1 + d(x, z) + d(z, y)
+
d(z, y)
1 + d(x, z) + d(z, y)
≤
d(x, z)
1 + d(x, z)
+
d(z, y)
1 + d(z, y)
.
So done.
We can also extend the notion of Lipschitz equivalence to metric spaces.
Definition (Lipschitz equivalent metrics). Metrics
d, d
′
on a set
X
are said to
be Lipschitz equivalent if there are (positive) constants A, B such that
Ad(x, y) ≤ d
′
(x, y) ≤ Bd(x, y)
for all x, y ∈ X.
Clearly, any Lipschitz equivalent norms give Lipschitz equivalent metrics. Any
metric coming from a norm in
R
n
is thus Lipschitz equivalent to the Euclidean
metric. We will later show that two equivalent norms induce the same topology.
In some sense, Lipschitz equivalent norms are indistinguishable.
Definition (Metric subspace). Given a metric space (
X, d
) and a subset
Y ⊆ X
,
the restriction
d|
Y ×Y
→ R
is a metric on
Y
. This is called the induced metric
or subspace metric.
Note that unlike vector subspaces, we do not require our subsets to have any
structure. We can take any subset of X and get a metric subspace.
Example. Any subspace of R
n
is a metric space with the Euclidean metric.
Definition (Convergence). Let (
X, d
) be a metric space. A sequence
x
n
∈ X
is
said to converge to x if d(x
n
, x) → 0 as a real sequence. In other words,
(∀ε)(∃K)(∀k > K) d(x
k
, x) < ε.
Alternatively, this says that given any
ε
, for sufficiently large
k
, we get
x
k
∈
B
ε
(x).
Again, B
r
(a) is the open ball centered at a with radius r, defined as
B
r
(a) = {x ∈ X : d(x, a) < r}.
Proposition. The limit of a convergent sequence is unique.
Proof. Same as that of normed spaces.
Note that notions such as convergence, open and closed subsets and continuity
of mappings all make sense in an even more general setting called topological
spaces. However, in this setting, limits of convergent sequences can fail to be
unique. We will not worry ourselves about these since we will just focus on
metric spaces.