In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 cal ...
s and
series of real numbers, and
real functions. Some particular properties of real-valued sequences and functions that real analysis studies include
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
,
limits,
continuity,
smoothness,
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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, 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 for ...
s and their functions.
Scope
Construction of the real numbers
The theorems of real analysis rely on the properties of the (established)
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
system. The real number system consists of an
uncountable set
In mathematics, an uncountable set, informally, 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 number is larger t ...
(
), 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, a binary operation ...
s denoted and
, and a
total order denoted . The operations make the real numbers a
field, and, along with the order, an
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
. The real number system is the unique ''
complete ordered field'', in the sense that any other complete ordered field is
isomorphic to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') 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 mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
, 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 and the
mean value theorem.
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, 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 x 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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. The topology induced by metric
turns out to be identical to the standard topology induced by order
. Theorems like the
intermediate value theorem 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 is a
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
,
totally ordered set.
The domain is usually taken to be the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 (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