Real analysis

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences 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, differentiability and integrability.

Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

Scope

Construction of the real numbers

The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set ($\mathbb$), together with two binary operations denoted and , and an order 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 ordered field'', in the sense that any other complete ordered field is isomorphic 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 $\mathbb$) 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 $\mathbb$ 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 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, the real numbers has a ''standard topology'', which is the order topology induced by order $<$. Alternatively, by defining the ''metric'' or ''distance function'' $d:\mathbb\times\mathbb\to\mathbb_$ using the absolute value function as the real numbers become the prototypical example of a metric space. The topology induced by metric $d$ 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 $\mathbb$ 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, totally ordered set. The domain is usually taken to be the natural numbers, 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 $a : \N \to \R : n \mapsto a_n$. Each $a(n) = a_n$ 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:
$(a_n) = (a_n)_=(a_1, a_2, a_3, \dots) .$
A sequence that tends to a limit (i.e., $\lim_ a_n$ exists) is said to be convergent; otherwise it is divergent. (''See the section on limits and convergence for details.'') A real-valued sequence $(a_n)$ is ''bounded'' if there exists $M\in\R$ such that , a_n, for all $n\in\mathbb$. A real-valued sequence $(a_n)$ is ''monotonically increasing'' or ''decreasing'' if
$a_1 \leq a_2 \leq a_3 \leq \cdots$
or
$a_1 \geq a_2 \geq a_3 \geq \cdots$
holds, respectively. If either holds, the sequence is said to be ''monotonic''. The monotonicity is ''strict'' if the chained inequalities still hold with $\leq$ or $\geq$ replaced by < or >.

Given a sequence $(a_n)$, another sequence $(b_k)$ is a ''subsequence'' of $(a_n)$ if $b_k=a_$ for all positive integers $k$ and $(n_k)$ is a strictly increasing sequence of natural numbers.

Limits and convergence

Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value. (This value can include the symbols $\pm\infty$ when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a limit is fundamental to calculus (and mathematical analysis in general) and its formal definition is used in turn to define notions like continuity, derivatives, and integrals. (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.)

The concept of limit was informally introduced for functions by Newton and Leibniz, at the end of the 17th century, for building infinitesimal calculus. For sequences, the concept was introduced by Cauchy, and made rigorous, at the end of the 19th century by Bolzano and Weierstrass, who gave the modern ε-δ definition, which follows.

Definition. Let $f$ be a real-valued function defined on We say that ''$f(x)$ tends to $L$ as $x$ approaches $x_0$'', or that ''the limit of $f(x)$ as $x$ approaches $x_0$ is $L$'' if, for any $\varepsilon>0$, there exists $\delta>0$ such that for all $x\in E$, $0 < , x - x_0, < \delta$ implies that $, f(x) - L, < \varepsilon$. We write this symbolically as
$f(x)\to L\ \ \text\ \ x\to x_0 ,$ or as $\lim_ f(x) = L .$
Intuitively, this definition can be thought of in the following way: We say that $f(x)\to L$ as $x\to x_0$, when, given any positive number $\varepsilon$, no matter how small, we can always find a $\delta$, such that we can guarantee that $f(x)$ and $L$ are less than $\varepsilon$ apart, as long as $x$ (in the domain of $f$) is a real number that is less than $\delta$ away from $x_0$ but distinct from $x_0$. The purpose of the last stipulation, which corresponds to the condition $0<, x-x_0,$ in the definition, is to ensure that $\lim_ f(x)=L$ does not imply anything about the value of $f(x_0)$ itself. Actually, $x_0$ does not even need to be in the domain of $f$ in order for $\lim_ f(x)$ to exist.

In a slightly different but related context, the concept of a limit applies to the behavior of a sequence $(a_n)$ when $n$ becomes large.

Definition. Let $(a_n)$ be a real-valued sequence. We say that $(a_n)$ ''converges to'' $a$ if, for any $\varepsilon > 0$, there exists a natural number $N$ such that $n\geq N$ implies that $, a-a_n, < \varepsilon$. We write this symbolically as
$a_n \to a\ \ \text\ \ n \to \infty ,$or as$\lim_ a_n = a ;$
if $(a_n)$ fails to converge, we say that $(a_n)$ ''diverges''.

Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence $(a_n)$ and term $a_n$ by function $f$ and value $f(x)$ and natural numbers $N$ and $n$ by real numbers $M$ and $x$, respectively) yields the definition of the ''limit of $f(x)$ as $x$ increases without bound'', notated $\lim_ f(x)$. Reversing the inequality $x\geq M$ to $x \leq M$ gives the corresponding definition of the limit of $f(x)$ as $x$ ''decreases'' ''without bound'',

Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant. In these cases, the concept of a Cauchy sequence is useful.

Definition. Let $(a_n)$ be a real-valued sequence. We say that $(a_n)$ is a ''Cauchy sequence'' if, for any $\varepsilon > 0$, there exists a natural number $N$ such that $m,n\geq N$ implies that $, a_m-a_n, < \varepsilon$.

It can be shown that a real-valued sequence is Cauchy if and only if it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, $(\R, , \cdot, )$, is a ''complete metric space''. In a general metric space, however, a Cauchy sequence need not converge.

In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.

Uniform and pointwise convergence for sequences of functions

In addition to sequences of numbers, one may also speak of ''sequences of functions'' ''on'' $E\subset \mathbb$, that is, infinite, ordered families of functions $f_n:E\to\mathbb$, denoted $(f_n)_^\infty$, and their convergence properties. However, in the case of sequences of functions, there are two kinds of convergence, known as ''pointwise convergence'' and ''uniform convergence'', that need to be distinguished.

Roughly speaking, pointwise convergence of functions $f_n$ to a limiting function $f:E\to\mathbb$, denoted $f_n \rightarrow f$, simply means that given any $x\in E$, $f_n(x)\to f(x)$ as $n\to\infty$. In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely. Uniform convergence requires members of the family of functions, $f_n$, to fall within some error $\varepsilon > 0$ of $f$ for ''every value of $x\in E$'', whenever $n\geq N$, for some integer $N$. For a family of functions to uniformly converge, sometimes denoted $f_n\rightrightarrows f$, such a value of $N$ must exist for any $\varepsilon>0$ given, no matter how small. Intuitively, we can visualize this situation by imagining that, for a large enough $N$, the functions $f_N, f_, f_,\ldots$ are all confined within a 'tube' of width $2\varepsilon$ about $f$ (that is, between $f - \varepsilon$ and $f+\varepsilon$) ''for every value in their domain'' $E$.

The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence. For example, a sequence of continuous functions (see below) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise. Karl Weierstrass is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.

Compactness

Compactness is a concept from general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being ''closed'' and ''bounded''. (In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.) Briefly, a closed set contains all of its boundary points, while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number. In $\mathbb$, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, closed intervals, and their finite unions. However, this list is not exhaustive; for instance, the set $\\cup \\$ is a compact set; the Cantor ternary set \mathcal\subset ,1/math> is another example of a compact set. On the other hand, the set $\$ is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set. The set Heine-Borel theorem.

A more general definition that applies to all metric spaces uses the notion of a subsequence (see above).

Definition. A set $E$ in a metric space is compact if every sequence in $E$ has a convergent subsequence.

This particular property is known as ''subsequential compactness''. In $\mathbb$, 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 $\mathbb$ as special cases). In brief, a collection of open sets $U_$ is said to be an ''open cover'' of set $X$ if the union of these sets is a superset of $X$. This open cover is said to have a ''finite subcover'' if a finite subcollection of the $U_$ could be found that also covers $X$.

Definition. A set $X$ in a topological space is compact if every open cover of $X$ 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.

Continuity

A function from the set of real numbers to the real numbers can be represented by a graph