In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the branch of real analysis studies the behavior of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s,
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 and
series of real numbers, and
real functions. Some particular properties of real-valued sequences and functions that real analysis studies include
convergence,
limits,
continuity,
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
,
differentiability and
integrability.
Real analysis is distinguished from
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, which deals with the study 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 fo ...
s and their functions.
Scope
Construction of the real numbers
The theorems of real analysis rely on the properties of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
system, which must be established. The real number system consists of an
uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
(
), together with two
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s denoted and , and an
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
denoted . The operations make the real numbers a
field, and, along with the order, an
ordered field. The real number system is the unique ''
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
ordered field'', in the sense that any other complete ordered field is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to it. Intuitively, completeness means that there are no 'gaps' in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers
) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the ''least upper bound property'' (see below).
Order properties of the real numbers
The real numbers have various
lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an
ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is
total, and the real numbers have the
least upper bound property:
''Every nonempty subset of that has an upper bound has a least upper bound that is also a real number.''
These
order-theoretic properties lead to a number of fundamental results in real analysis, such as the
monotone convergence theorem, the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two impor ...
and the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
.
However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and
operator theory generalize properties of the real numbers – such generalizations include the theories of
Riesz spaces and
positive operators. Also, mathematicians consider
real and
imaginary parts of complex sequences, or by
pointwise evaluation of
operator sequences.
Topological properties of the real numbers
Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, the real numbers has a ''standard topology'', which is the
order topology induced by order
. Alternatively, by defining the ''metric'' or ''distance function''
using the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function as the real numbers become the prototypical example of a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. The topology induced by metric
turns out to be identical to the standard topology induced by order
. Theorems like the
intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two impor ...
that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in
only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.
Sequences
A ''sequence'' is a
function whose
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
is a
countable
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 numbers ...
,
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
set. The domain is usually taken to be the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s,
although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices.
Of interest in real analysis, a ''real-valued sequence'', here indexed by the natural numbers, is a map
. Each
is referred to as a ''term'' (or, less commonly, an ''element'') of the sequence. A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses:
A sequence that tends to 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 ...
(i.e.,
exists) is said to be convergent; otherwise it is divergent. (''See the section on limits and convergence for details.'') A real-valued sequence
is ''bounded'' if there exists
such that
.
A_more_general_definition_that_applies_to_all_metric_spaces_uses_the_notion_of_a_subsequence_(see_above).
Definition._A_set_
_has_a_convergent_subsequence.
This_particular_property_is_known_as_''subsequential_compactness''._In_
,_a_set_is_subsequentially_compact_if_and_only_if_it_is_closed_and_bounded,_making_this_definition_equivalent_to_the_one_given_above.__Subsequential_compactness_is_equivalent_to_the_definition_of_compactness_based_on_subcovers_for_metric_spaces,_but_not_for_topological_spaces_in_general.
The_most_general_definition_of_compactness_relies_on_the_notion_of_''open_covers''_and_''subcovers'',_which_is_applicable_to_topological_spaces_(and_thus_to_metric_spaces_and_
.__This_open_cover_is_said_to_have_a_''finite_subcover''_if_a_finite_subcollection_of_the_
_has_a_finite_subcover.
Compact_sets_are_well-behaved_with_respect_to_properties_like_convergence_and_continuity._For_instance,_any_Cauchy_sequence_in_a_compact_metric_space_is_convergent._As_another_example,_the_image_of_a_compact_metric_space_under_a_continuous_map_is_also_compact.