In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a
metric space is called complete (or a Cauchy space) if every
Cauchy sequence of points in has a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of
rational numbers is not complete, because e.g.
is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below.
Definition
Cauchy sequence
A sequence
in a
metric space is called Cauchy if for every positive
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
there is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
such that for all positive integers
Complete space
A metric space
is complete if any of the following equivalent conditions are satisfied:
:#Every
Cauchy sequence of points in
has a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
that is also in
:#Every Cauchy sequence in
converges in
(that is, to some point of
).
:#Every decreasing sequence of
non-empty closed
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of
with
diameters tending to 0, has a non-empty
intersection: if
is closed and non-empty,
for every
and
then there is a point
common to all sets
Examples
The space Q of
rational numbers, with the standard metric given by the
absolute value of the
difference, is not complete.
Consider for instance the sequence defined by
and
This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit
then by solving
necessarily
yet no rational number has this property.
However, considered as a sequence of
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, it does converge to the
irrational number .
The
open interval , again with the absolute value metric, is not complete either.
The sequence defined by is Cauchy, but does not have a limit in the given space.
However the
closed interval
is complete; for example the given sequence does have a limit in this interval and the limit is zero.
The space R of real numbers and the space C of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s (with the metric given by the absolute value) are complete, and so is
Euclidean space R
''n'', with the
usual distance metric.
In contrast, infinite-dimensional
normed vector spaces may or may not be complete; those that are complete are
Banach spaces.
The space C of
continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when ...
.
However, the supremum norm does not give a norm on the space C of continuous functions on , for it may contain unbounded functions.
Instead, with the topology of
compact convergence, C can be given the structure of a
Fréchet space: a
locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.
The space Q
''p'' of
''p''-adic numbers is complete for any
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
This space completes Q with the ''p''-adic metric in the same way that R completes Q with the usual metric.
If
is an arbitrary set, then the set of all
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s in
becomes a complete metric space if we define the distance between the sequences
and
to be
where
is the smallest index for which
is
distinct from
or
if there is no such index.
This space is
homeomorphic to the
product of a
countable number of copies of the
discrete space
Riemannian manifolds which are complete are called
geodesic manifolds; completeness follows from the
Hopf–Rinow theorem.
Some theorems
Every
compact metric space is complete, though complete spaces need not be compact. In fact, a metric space is compact
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
it is complete and
totally bounded. This is a generalization of the
Heine–Borel theorem, which states that any closed and bounded subspace
of is compact and therefore complete.
Let
be a complete metric space. If
is a closed set, then
is also complete.
Let
be a metric space. If
is a complete subspace, then
is also closed.
If
is a
set and
is a complete metric space, then the set
of all
bounded functions from to
is a complete metric space. Here we define the distance in
in terms of the distance in
with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when ...
If
is a
topological space and
is a complete metric space, then the set
consisting of all
continuous bounded functions
is a closed subspace of
and hence also complete.
The
Baire category theorem says that every complete metric space is a
Baire space. That is, the
union of
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
nowhere dense subsets of the space has empty
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
.
The
Banach fixed-point theorem states that a contraction mapping on a complete metric space admits a fixed point. The fixed-point theorem is often used to prove the
inverse function theorem on complete metric spaces such as Banach spaces.
Completion
For any metric space ''M'', it is possible to construct a complete metric space ''M′'' (which is also denoted as
), which contains ''M'' as a
dense subspace. It has the following
universal property: if ''N'' is any complete metric space and ''f'' is any
uniformly continuous function from ''M'' to ''N'', then there exists a
unique uniformly continuous function ''f′'' from ''M′'' to ''N'' that extends ''f''. The space ''M is determined
up to isometry by this property (among all complete metric spaces isometrically containing ''M''), and is called the ''completion'' of ''M''.
The completion of ''M'' can be constructed as a set of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of Cauchy sequences in ''M''. For any two Cauchy sequences
and
in ''M'', we may define their distance as
(This limit exists because the real numbers are complete.) This is only a
pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of ''M''. The original space is embedded in this space via the identification of an element ''x'' of ''M with the equivalence class of sequences in ''M'' converging to ''x'' (i.e., the equivalence class containing the sequence with constant value ''x''). This defines an
isometry onto a dense subspace, as required. Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.
Cantor's construction of the real numbers is similar to the above construction; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances. The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction. Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a
field that has the rational numbers as a subfield. This field is complete, admits a natural
total ordering, and is the unique totally ordered complete field (up to isomorphism). It is ''defined'' as the field of real numbers (see also
Construction of the real numbers for more details). One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that "ought" to have a given real limit is identified with that real number. The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class.
For a prime
the
-adic numbers arise by completing the rational numbers with respect to a different metric.
If the earlier completion procedure is applied to a
normed vector space, the result is a
Banach space containing the original space as a dense subspace, and if it is applied to an
inner product space, the result is a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
containing the original space as a dense subspace.
Topologically complete spaces
Completeness is a property of the ''metric'' and not of the ''
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
'', meaning that a complete metric space can be
homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval , which is not complete.
In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
one considers ''
completely metrizable spaces'', spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the
Baire category theorem is purely topological, it applies to these spaces as well.
Completely metrizable spaces are often called ''topologically complete''. However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section
Alternatives and generalizations). Indeed, some authors use the term ''topologically complete'' for a wider class of topological spaces, the
completely uniformizable spaces.
[Kelley, Problem 6.L, p. 208]
A topological space homeomorphic to a
separable complete metric space is called a
Polish space.
Alternatives and generalizations
Since
Cauchy sequences can also be defined in general
topological groups, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure. This is most often seen in the context of
topological vector spaces, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points
and
is gauged not by a real number
via the metric
in the comparison
but by an open neighbourhood
of
via subtraction in the comparison
A common generalisation of these definitions can be found in the context of a
uniform space, where an
entourage is a set of all pairs of points that are at no more than a particular "distance" from each other.
It is also possible to replace Cauchy ''sequences'' in the definition of completeness by Cauchy ''
nets'' or
Cauchy filters. If every Cauchy net (or equivalently every Cauchy filter) has a limit in
then
is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is
Cauchy spaces; these too have a notion of completeness and completion just like uniform spaces.
See also
*
*
*
*
*
*
Notes
References
*
*
Kreyszig, Erwin, ''Introductory functional analysis with applications'' (Wiley, New York, 1978).
*
Lang, Serge, "Real and Functional Analysis"
*
{{DEFAULTSORT:Complete Metric Space
Metric geometry
Topology
Uniform spaces