26.1
Metric Spaces and Normed Vector Spaces
This chapter contains a review of basic topological concepts. First, metric spaces are defined.
Next, normed vector spaces are defined. Closed and open sets are defined, and their basic
properties are stated. The general concept of a topological space is defined. The closure and
the interior of a subset are defined. The subspace topology and the product topology are
defined. Continuous maps and homeomorphisms are defined. Limits of seqences are defined.
Continuous linear maps and multilinear maps are defined and studied briefly. The chapter
ends with the definition of a normed affine space.
Most spaces considered in this book have a topological structure given by a metric or a
norm, and we first review these notions. We begin with metric spaces. Recall that R+ =
{x ∈ R | x ≥ 0}.
Definition 26.1. A metric space is a set E together with a function d : E × E → R+,
called a metric, or distance, assigning a nonnegative real number d(x, y) to any two points
x, y ∈ E, and satisfying the following conditions for all x, y, z ∈ E:
(D1) d(x, y) = d(y, x).
(symmetry)
(D2) d(x, y) ≥ 0, and d(x, y) = 0 iff x = y.
(positivity)
(D3) d(x, z) ≤ d(x, y) + d(y, z).
(triangle inequality)
Geometrically, condition (D3) expresses the fact that in a triangle with vertices x, y, z,
the length of any side is bounded by the sum of the lengths of the other two sides. From
(D3), we immediately get
|d(x, y) − d(y, z)| ≤ d(x, z).
Let us give some examples of metric spaces. Recall that the absolute value |x| of a real
number x ∈ R is defined such that |x| = x if x ≥ 0, |x| = −x if x < 0, and for a complex
√
number x = a + ib, by |x| = a2 + b2.
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CHAPTER 26. TOPOLOGY
Example 26.1.
1. Let E = R, and d(x, y) = |x − y|, the absolute value of x − y. This is the so-called
natural metric on R.
2. Let E = n
n
R (or E = C ). We have the Euclidean metric
1
d
2
2(x, y) =
|x1 − y1|2 + · · · + |xn − yn|2
,
the distance between the points (x1, . . . , xn) and (y1, . . . , yn).
3. For every set E, we can define the discrete metric, defined such that d(x, y) = 1 iff
x = y, and d(x, x) = 0.
4. For any a, b ∈ R such that a < b, we define the following sets:
[a, b] = {x ∈ R | a ≤ x ≤ b},
(closed interval)
]a, b[ = {x ∈ R | a < x < b},
(open interval)
[a, b[ = {x ∈ R | a ≤ x < b},
(interval closed on the left, open on the right)
]a, b] = {x ∈ R | a < x ≤ b},
(interval open on the left, closed on the right)
Let E = [a, b], and d(x, y) = |x − y|. Then, ([a, b], d) is a metric space.
We will need to define the notion of proximity in order to define convergence of limits
and continuity of functions. For this, we introduce some standard “small neighborhoods.”
Definition 26.2. Given a metric space E with metric d, for every a ∈ E, for every ρ ∈ R,
with ρ > 0, the set
B(a, ρ) = {x ∈ E | d(a, x) ≤ ρ}
is called the closed ball of center a and radius ρ, the set
B0(a, ρ) = {x ∈ E | d(a, x) < ρ}
is called the open ball of center a and radius ρ, and the set
S(a, ρ) = {x ∈ E | d(a, x) = ρ}
is called the sphere of center a and radius ρ. It should be noted that ρ is finite (i.e., not
+∞). A subset X of a metric space E is bounded if there is a closed ball B(a, ρ) such that
X ⊆ B(a, ρ).
Clearly, B(a, ρ) = B0(a, ρ) ∪ S(a, ρ).
26.1. METRIC SPACES AND NORMED VECTOR SPACES
735
Example 26.2.
1. In E = R with the distance |x − y|, an open ball of center a and radius ρ is the open
interval ]a − ρ, a + ρ[.
2. In E =
2
R with the Euclidean metric, an open ball of center a and radius ρ is the set
of points inside the disk of center a and radius ρ, excluding the boundary points on
the circle.
3. In E =
3
R with the Euclidean metric, an open ball of center a and radius ρ is the set
of points inside the sphere of center a and radius ρ, excluding the boundary points on
the sphere.
One should be aware that intuition can be misleading in forming a geometric image of a
closed (or open) ball. For example, if d is the discrete metric, a closed ball of center a and
radius ρ < 1 consists only of its center a, and a closed ball of center a and radius ρ ≥ 1
consists of the entire space!
If E = [a, b], and d(x, y) = |x − y|, as in Example 26.1, an open ball B0(a, ρ), with
ρ < b − a, is in fact the interval [a, a + ρ[, which is closed on the left.
We now consider a very important special case of metric spaces, normed vector spaces.
Normed vector spaces have already been defined in Chapter 7 (Definition 7.1) but for the
reader’s convenience we repeat the definition.
Definition 26.3. Let E be a vector space over a field K, where K is either the field R of
reals, or the field C of complex numbers. A norm on E is a function
: E → R+, assigning
a nonnegative real number u to any vector u ∈ E, and satisfying the following conditions
for all x, y, z ∈ E:
(N1) x ≥ 0, and x = 0 iff x = 0.
(positivity)
(N2) λx = |λ| x .
(scaling)
(N3) x + y ≤ x + y .
(triangle inequality)
A vector space E together with a norm
is called a normed vector space.
From (N3), we easily get
| x − y | ≤ x − y .
Given a normed vector space E, if we define d such that
d(x, y) = x − y ,
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CHAPTER 26. TOPOLOGY
it is easily seen that d is a metric. Thus, every normed vector space is immediately a metric
space. Note that the metric associated with a norm is invariant under translation, that is,
d(x + u, y + u) = d(x, y).
For this reason, we can restrict ourselves to open or closed balls of center 0.
Examples of normed vector spaces were given in Example 7.1. We repeat the most
important examples.
Example 26.3. Let E =
n
n
R
(or E = C ). There are three standard norms. For every
(x1, . . . , xn) ∈ E, we have the norm x 1, defined such that,
x 1 = |x1| + · · · + |xn|,
we have the Euclidean norm x 2, defined such that,
1
x
2
2 =
|x1|2 + · · · + |xn|2
,
and the sup-norm x ∞, defined such that,
x ∞ = max{|xi| | 1 ≤ i ≤ n}.
More generally, we define the p-norm (for p ≥ 1) by
x p = (|x1|p + · · · + |xn|p)1/p.
We proved in Proposition 7.1 that the p-norms are indeed norms. One should work out
what are the open balls in
2
R for
1 and
∞.
In a normed vector space, we define a closed ball or an open ball of radius ρ as a closed
ball or an open ball of center 0. We may use the notation B(ρ) and B0(ρ).
We will now define the crucial notions of open sets and closed sets, and of a topological
space.
Definition 26.4. Let E be a metric space with metric d. A subset U ⊆ E is an open
set in E if either U = ∅, or for every a ∈ U, there is some open ball B0(a, ρ) such that,
B0(a, ρ) ⊆ U.1 A subset F ⊆ E is a closed set in E if its complement E − F is open in E.
The set E itself is open, since for every a ∈ E, every open ball of center a is contained in
E. In E = n
R , given n intervals [ai, bi], with ai < bi, it is easy to show that the open n-cube
{(x1, . . . , xn) ∈ E | ai < xi < bi, 1 ≤ i ≤ n}
1Recall that ρ > 0.
26.2. TOPOLOGICAL SPACES
737
is an open set. In fact, it is possible to find a metric for which such open n-cubes are open
balls! Similarly, we can define the closed n-cube
{(x1, . . . , xn) ∈ E | ai ≤ xi ≤ bi, 1 ≤ i ≤ n},
which is a closed set.
The open sets satisfy some important properties that lead to the definition of a topological
space.
Proposition 26.1. Given a metric space E with metric d, the family O of all open sets
defined in Definition 26.4 satisfies the following properties:
(O1) For every finite family (Ui)1≤i≤n of sets Ui ∈ O, we have U1 ∩ · · · ∩ Un ∈ O, i.e., O is
closed under finite intersections.
(O2) For every arbitrary family (Ui)i∈I of sets Ui ∈ O, we have
U
i∈I
i ∈ O, i.e., O is closed
under arbitrary unions.
(O3) ∅ ∈ O, and E ∈ O, i.e., ∅ and E belong to O.
Furthermore, for any two distinct points a = b in E, there exist two open sets Ua and Ub
such that, a ∈ Ua, b ∈ Ub, and Ua ∩ Ub = ∅.
Proof. It is straightforward. For the last point, letting ρ = d(a, b)/3 (in fact ρ = d(a, b)/2
works too), we can pick Ua = B0(a, ρ) and Ub = B0(b, ρ). By the triangle inequality, we
must have Ua ∩ Ub = ∅.
The above proposition leads to the very general concept of a topological space.
One should be careful that, in general, the family of open sets is not closed under infinite
intersections. For example, in R under the metric |x − y|, letting Un =] − 1/n, +1/n[,
each Un is open, but
U
n
n = {0}, which is not open.
26.2
Topological Spaces
Motivated by Proposition 26.1, a topological space is defined in terms of a family of sets
satisfing the properties of open sets stated in that proposition.
Definition 26.5. Given a set E, a topology on E (or a topological structure on E), is defined
as a family O of subsets of E called open sets, and satisfying the following three properties:
(1) For every finite family (Ui)1≤i≤n of sets Ui ∈ O, we have U1 ∩ · · · ∩ Un ∈ O, i.e., O is
closed under finite intersections.
(2) For every arbitrary family (Ui)i∈I of sets Ui ∈ O, we have
U
i∈I
i ∈ O, i.e., O is closed
under arbitrary unions.
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CHAPTER 26. TOPOLOGY
(3) ∅ ∈ O, and E ∈ O, i.e., ∅ and E belong to O.
A set E together with a topology O on E is called a topological space. Given a topological
space (E, O), a subset F of E is a closed set if F = E − U for some open set U ∈ O, i.e., F
is the complement of some open set.
It is possible that an open set is also a closed set. For example, ∅ and E are both open
and closed. When a topological space contains a proper nonempty subset U which is
both open and closed, the space E is said to be disconnected .
A topological space (E, O) is said to satisfy the Hausdorff separation axiom (or T2-
separation axiom) if for any two distinct points a = b in E, there exist two open sets Ua and
Ub such that, a ∈ Ua, b ∈ Ub, and Ua ∩ Ub = ∅. When the T2-separation axiom is satisfied,
we also say that (E, O) is a Hausdorff space.
As shown by Proposition 26.1, any metric space is a topological Hausdorff space, the
family of open sets being in fact the family of arbitrary unions of open balls. Similarly,
any normed vector space is a topological Hausdorff space, the family of open sets being the
family of arbitrary unions of open balls. The topology O consisting of all subsets of E is
called the discrete topology.
Remark: Most (if not all) spaces used in analysis are Hausdorff spaces. Intuitively, the
Hausdorff separation axiom says that there are enough “small” open sets. Without this
axiom, some counter-intuitive behaviors may arise. For example, a sequence may have more
than one limit point (or a compact set may not be closed). Nevertheless, non-Hausdorff
topological spaces arise naturally in algebraic geometry. But even there, some substitute for
separation is used.
One of the reasons why topological spaces are important is that the definition of a topol-
ogy only involves a certain family O of sets, and not how such family is generated from
a metric or a norm. For example, different metrics or different norms can define the same
family of open sets. Many topological properties only depend on the family O and not on
the specific metric or norm. But the fact that a topology is definable from a metric or a
norm is important, because it usually implies nice properties of a space. All our examples
will be spaces whose topology is defined by a metric or a norm.
By taking complements, we can state properties of the closed sets dual to those of Defi-
nition 26.5. Thus, ∅ and E are closed sets, and the closed sets are closed under finite unions
and arbitrary intersections.
It is also worth noting that the Hausdorff separation axiom implies that for every a ∈ E,
the set {a} is closed. Indeed, if x ∈ E − {a}, then x = a, and so there exist open sets Ua
and Ux such that a ∈ Ua, x ∈ Ux, and Ua ∩ Ux = ∅. Thus, for every x ∈ E − {a}, there is an
open set Ux containing x and contained in E − {a}, showing by (O3) that E − {a} is open,
and thus that the set {a} is closed.
26.2. TOPOLOGICAL SPACES
739
Given a topological space (E, O), given any subset A of E, since E ∈ O and E is a closed
set, the family CA = {F | A ⊆ F, F a closed set} of closed sets containing A is nonempty,
and since any arbitrary intersection of closed sets is a closed set, the intersection
CA of
the sets in the family CA is the smallest closed set containing A. By a similar reasoning, the
union of all the open subsets contained in A is the largest open set contained in A.
Definition 26.6. Given a topological space (E, O), given any subset A of E, the smallest
closed set containing A is denoted by A, and is called the closure, or adherence of A. A
subset A of E is dense in E if A = E. The largest open set contained in A is denoted by
◦
A, and is called the interior of A. The set Fr A = A ∩ E − A is called the boundary (or
frontier) of A. We also denote the boundary of A by ∂A.
Remark: The notation A for the closure of a subset A of E is somewhat unfortunate,
since A is often used to denote the set complement of A in E. Still, we prefer it to more
cumbersome notations such as clo(A), and we denote the complement of A in E by E − A
(or sometimes, Ac).
By definition, it is clear that a subset A of E is closed iff A = A. The set Q of rationals
◦
◦
is dense in R. It is easily shown that A = A ∪ ∂A and A ∩ ∂A = ∅. Another useful
characterization of A is given by the following proposition.
Proposition 26.2. Given a topological space (E, O), given any subset A of E, the closure
A of A is the set of all points x ∈ E such that for every open set U containing x, then
U ∩ A = ∅.
Proof. If A = ∅, since ∅ is closed, the proposition holds trivially. Thus, assume that A = ∅.
First, assume that x ∈ A. Let U be any open set such that x ∈ U. If U ∩ A = ∅, since U is
open, then E − U is a closed set containing A, and since A is the intersection of all closed
sets containing A, we must have x ∈ E − U, which is impossible. Conversely, assume that
x ∈ E is a point such that for every open set U containing x, then U ∩ A = ∅. Let F be
any closed subset containing A. If x /
∈ F , since F is closed, then U = E − F is an open set
such that x ∈ U, and U ∩ A = ∅, a contradiction. Thus, we have x ∈ F for every closed set
containing A, that is, x ∈ A.
Often, it is necessary to consider a subset A of a topological space E, and to view the
subset A as a topological space. The following proposition shows how to define a topology
on a subset.
Proposition 26.3. Given a topological space (E, O), given any subset A of E, let
U = {U ∩ A | U ∈ O}
be the family of all subsets of A obtained as the intersection of any open set in O with A.
The following properties hold.
740
CHAPTER 26. TOPOLOGY
(1) The space (A, U) is a topological space.
(2) If E is a metric space with metric d, then the restriction dA : A × A → R+ of the
metric d to A defines a metric space. Furthermore, the topology induced by the metric
dA agrees with the topology defined by U, as above.
Proof. Left as an exercise.
Proposition 26.3 suggests the following definition.
Definition 26.7. Given a topological space (E, O), given any subset A of E, the subspace
topology on A induced by O is the family U of open sets defined such that
U = {U ∩ A | U ∈ O}
is the family of all subsets of A obtained as the intersection of any open set in O with A.
We say that (A, U) has the subspace topology. If (E, d) is a metric space, the restriction
dA : A × A → R+ of the metric d to A is called the subspace metric.
For example, if E =
n
R and d is the Euclidean metric, we obtain the subspace topology
on the closed n-cube
{(x1, . . . , xn) ∈ E | ai ≤ xi ≤ bi, 1 ≤ i ≤ n}.
One should realize that every open set U ∈ O which is entirely contained in A is also in
the family U, but U may contain open sets that are not in O. For example, if E = R
with |x − y|, and A = [a, b], then sets of the form [a, c[, with a < c < b belong to U, but they
are not open sets for R under |x − y|. However, there is agreement in the following situation.
Proposition 26.4. Given a topological space (E, O), given any subset A of E, if U is the
subspace topology, then the following properties hold.
(1) If A is an open set A ∈ O, then every open set U ∈ U is an open set U ∈ O.
(2) If A is a closed set in E, then every closed set w.r.t. the subspace topology is a closed
set w.r.t. O.
Proof. Left as an exercise.
The concept of product topology is also useful. We have the following proposition.
Proposition 26.5. Given n topological spaces (Ei, Oi), let B be the family of subsets of
E1 × · · · × En defined as follows:
B = {U1 × · · · × Un | Ui ∈ Oi, 1 ≤ i ≤ n},
and let P be the family consisting of arbitrary unions of sets in B, including ∅. Then, P is
a topology on E1 × · · · × En.
26.2. TOPOLOGICAL SPACES
741
Proof. Left as an exercise.
Definition 26.8. Given n topological spaces (Ei, Oi), the product topology on E1 × · · · × En
is the family P of subsets of E1 × · · · × En defined as follows: if
B = {U1 × · · · × Un | Ui ∈ Oi, 1 ≤ i ≤ n},
then P is the family consisting of arbitrary unions of sets in B, including ∅.
If each (Ei,
i) is a normed vector space, there are three natural norms that can be
defined on E1 × · · · × En:
(x1, . . . , xn) 1 = x1 1 + · · · + xn n,
1
(x
2
2
2
1, . . . , xn) 2 =
x1 1 + · · · + xn n
,
(x1, . . . , xn) ∞ = max { x1 1, . . . , xn n} .
It is easy to show that they all define the same topology, which is the product topology.
It can also be verified that when Ei = R, with the standard topology induced by |x − y|, the
topology product on
n
R is the standard topology induced by the Euclidean norm.
Definition 26.9. Two metrics d1 and d2 on a space E are equivalent if they induce the same
topology O on E (i.e., they define the same family O of open sets). Similarly, two norms
1 and
2 on a space E are equivalent if they induce the same topology O on E.
Remark: Given a topological space (E, O), it is often useful, as in Proposition 26.5, to
define the topology O in terms of a subfamily B of subsets of E. We say that a family B of
subsets of E is a basis for the topology O, if B is a subset of O, and if every open set U in
O can be obtained as some union (possibly infinite) of sets in B (agreeing that the empty
union is the empty set).
It is immediately verified that if a family B = (Ui)i∈I is a basis for the topology of (E, O),
then E =
U
i∈I
i, and the intersection of any two sets Ui, Uj ∈ B is the union of some sets in
the family B (again, agreeing that the empty union is the empty set). Conversely, a family
B with these properties is the basis of the topology obtained by forming arbitrary unions of
sets in B.
A subbasis for O is a family S of subsets of E, such that the family B of all finite
intersections of sets in S (including E itself, in case of the empty intersection) is a basis of
O.
The following proposition gives useful criteria for determining whether a family of open
subsets is a basis of a topological space.
Proposition 26.6. Given a topological space (E, O) and a family B of open subsets in O
the following properties hold:
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CHAPTER 26. TOPOLOGY
(1) The family B is a basis for the topology O iff for every open set U ∈ O and every
x ∈ U, there is some B ∈ B such that x ∈ B and B ⊆ U.
(2) The family B is a basis for the topology O iff
(a) For every x ∈ E, there is some B ∈ B such that x ∈ B.
(b) For any two open subsets, B1, B2 ∈ B, for every x ∈ E, if x ∈ B1 ∩ B2, then there
is some B3 ∈ B such that x ∈ B3 and B3 ⊆ B1 ∩ B2.
We now consider the fundamental property of continuity.
26.3
Continuous Functions, Limits
Definition 26.10. Let (E, OE) and (F, OF ) be topological spaces, and let f : E → F be a
function. For every a ∈ E, we say that f is continuous at a, if for every open set V ∈ OF
containing f (a), there is some open set U ∈ OE containing a, such that, f(U) ⊆ V . We say
that f is continuous if it is continuous at every a ∈ E.
Define a neighborhood of a ∈ E as any subset N of E containing some open set O ∈ O
such that a ∈ O. Now, if f is continuous at a and N is any neighborhood of f(a), there is
some open set V ⊆ N containing f(a), and since f is continuous at a, there is some open
set U containing a, such that f (U ) ⊆ V . Since V ⊆ N, the open set U is a subset of f−1(N)
containing a, and f −1(N ) is a neighborhood of a. Conversely, if f −1(N ) is a neighborhood
of a whenever N is any neighborhood of f (a), it is immediate that f is continuous at a. It
is easy to see that Definition 26.10 is equivalent to the following statements.
Proposition 26.7. Let (E, OE) and (F, OF ) be topological spaces, and let f : E → F be a
function. For every a ∈ E, the function f is continuous at a ∈ E iff for every neighborhood
N of f (a) ∈ F , then f−1(N) is a neighborhood of a. The function f is continuous on E iff
f −1(V ) is an open set in OE for every open set V ∈ OF .
If E and F are metric spaces defined by metrics d1 and d2, we can show easily that f is
continuous at a iff
for every > 0, there is some η > 0, such that, for every x ∈ E,
if d1(a, x) ≤ η, then d2(f(a), f(x)) ≤ .
Similarly, if E and F are normed vector spaces defined by norms
1 and
2, we can
show easily that f is continuous at a iff
for every > 0, there is some η > 0, such that, for every x ∈ E,
if x − a 1 ≤ η, then f(x) − f(a) 2 ≤ .
26.3. CONTINUOUS FUNCTIONS, LIMITS
743
It is worth noting that continuity is a topological notion, in the sense that equivalent
metrics (or equivalent norms) define exactly the same notion of continuity.
If (E, OE) and (F, OF ) are topological spaces, and f : E → F is a function, for every
nonempty subset A ⊆ E of E, we say that f is continuous on A if the restriction of f to A
is continuous with respect to (A, U) and (F, OF ), where U is the subspace topology induced
by OE on A.
Given a product E1×· · ·×En of topological spaces, as usual, we let πi : E1×· · ·×En → Ei
be the projection function such that, πi(x1, . . . , xn) = xi. It is immediately verified that each
πi is continuous.
Given a topological space (E, O), we say that a point a ∈ E is isolated if {a} is an open
set in O. Then, if (E, OE) and (F, OF ) are topological spaces, any function f :