Movatterモバイル変換

[0]ホーム

Jump to content

Real analysis

Edit links

From Wikipedia, the free encyclopedia

Not to be confused withReanalysis.

Mathematics of real numbers and real functions

This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Real analysis" – news ·newspapers ·books ·scholar ·JSTOR(October 2024) (Learn how and when to remove this message)

Part of a series on

Mathematics

Areas

Relationship with sciences

Mathematics Portal

Inmathematics, the branch ofreal analysis studies the behavior ofreal numbers,sequences andseries of real numbers, andreal functions.^[1] Some particular properties of real-valued sequences and functions that real analysis studies includeconvergence,limits,continuity,smoothness,differentiability andintegrability.

Real analysis is distinguished fromcomplex analysis, which deals with the study ofcomplex numbers and their functions.

Scope

[edit]

This section mayrequirecleanup to meet Wikipedia'squality standards. The specific problem is:This section goes too heavily into detail about each concept. It should just portray a brief overview in relation to the field of real analysis. Please helpimprove this section if you can.(June 2019) (Learn how and when to remove this message)

Construction of the real numbers

[edit]

Main article:Construction of the real numbers

The theorems of real analysis rely on the properties of thereal number system, which must be established. The real number system consists of anuncountable set ( $\mathbb {R}$ ), together with twobinary operations denoted+ and⋅, and atotal order denoted≤. The operations make the real numbers afield, and, along with the order, anordered field. The real number system is the uniquecomplete ordered field, in the sense that any other complete ordered field isisomorphic 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 $\mathbb {Q}$ ) 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 theleast upper bound property (see below).

Order properties of the real numbers

[edit]

The real numbers have variouslattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form anordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers istotal, and the real numbers have theleast upper bound property:

Every nonempty subset of $\mathbb {R}$ that has an upper bound has aleast upper bound that is also a real number.

Theseorder-theoretic properties lead to a number of fundamental results in real analysis, such as themonotone convergence theorem, theintermediate value theorem and themean 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 infunctional analysis andoperator theory generalize properties of the real numbers – such generalizations include the theories ofRiesz spaces andpositive operators. Also, mathematicians considerreal andimaginary parts of complex sequences, or bypointwise evaluation ofoperator sequences.^{[clarification needed]}

Topological properties of the real numbers

[edit]

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 atopological space, the real numbers has astandard topology, which is theorder topology induced by order $<$ . Alternatively, by defining themetric ordistance function $d:\mathbb {R} \times \mathbb {R} \to \mathbb {R} _{\geq 0}$ using theabsolute value function as $d(x,y)=|x-y|$ , the real numbers become the prototypical example of ametric space. The topology induced by metric $d {\displaystyle d}$ turns out to be identical to the standard topology induced by order $<$ . Theorems like theintermediate 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 $\mathbb {R}$ only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.

Sequences

[edit]

Main article:Sequence

Asequence is afunction whosedomain is acountable,totally ordered set.^[2] The domain is usually taken to be thenatural numbers,^[3] 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, areal-valued sequence, here indexed by the natural numbers, is a map $a:\mathbb {N} \to \mathbb {R} :n\mapsto a_{n}$ . Each $a(n)=a_{n}$ is referred to as aterm (or, less commonly, anelement) 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:^[4] $(a_{n})=(a_{n})_{n\in \mathbb {N} }=(a_{1},a_{2},a_{3},\dots ).$ A sequence that tends to alimit (i.e., ${\textstyle \lim _{n\to \infty }a_{n}}$ exists) is said to beconvergent; otherwise it isdivergent. (See the section on limits and convergence for details.) A real-valued sequence $(a_{n})$ isbounded if there exists $M\in \mathbb {R}$ such that $|a_{n}|<M$ for all $n\in \mathbb {N}$ . A real-valued sequence $(a_{n})$ ismonotonically increasing ordecreasing 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 bemonotonic. The monotonicity isstrict if the chained inequalities still hold with $\leq$ or $\geq$ replaced by < or >.

Given a sequence $(a_{n})$ , another sequence $(b_{k})$ is asubsequence of $(a_{n})$ if $b_{k}=a_{n_{k}}$ for all positive integers $k {\displaystyle k}$ and $(n_{k})$ is a strictly increasing sequence of natural numbers.

Limits and convergence

[edit]

Main article:Limit (mathematics)

Roughly speaking, alimit is the value that afunction or asequence "approaches" as the input or index approaches some value.^[5] (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 tocalculus (andmathematical analysis in general) and its formal definition is used in turn to define notions likecontinuity,derivatives, andintegrals. (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 byNewton andLeibniz, at the end of the 17th century, for buildinginfinitesimal calculus. For sequences, the concept was introduced byCauchy, and made rigorous, at the end of the 19th century byBolzano andWeierstrass, who gave the modernε-δ definition, which follows.

Definition. Let $f {\displaystyle f}$ be a real-valued function defined on $E\subset \mathbb {R}$ . We say that $f(x)$ tends to $L {\displaystyle L}$ as $x {\displaystyle x}$ approaches $x_{0}$ , or thatthe limit of $f(x)$ as $x {\displaystyle x}$ approaches $x_{0}$ is $L {\displaystyle L}$ 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 $f(x)\to L\ \ {\text{as}}\ \ x\to x_{0},$ or as $\lim _{x\to x_{0}}f(x)=L.$ 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 $L {\displaystyle L}$ are less than $\varepsilon$ apart, as long as $x {\displaystyle x}$ (in the domain of $f {\displaystyle f}$ ) 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 ${\textstyle \lim _{x\to x_{0}}f(x)=L}$ 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 $f {\displaystyle f}$ in order for ${\textstyle \lim _{x\to x_{0}}f(x)}$ to exist.

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

Definition. Let $(a_{n})$ be a real-valued sequence. We say that $(a_{n})$ converges to $a {\displaystyle a}$ if, for any $\varepsilon >0$ , there exists a natural number $N {\displaystyle N}$ such that $n\geq N$ implies that $|a-a_{n}|<\varepsilon$ . We write this symbolically as $a_{n}\to a\ \ {\text{as}}\ \ n\to \infty ,$ or as $\lim _{n\to \infty }a_{n}=a;$ 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 $f {\displaystyle f}$ and value $f(x)$ and natural numbers $N {\displaystyle N}$ and $n {\displaystyle n}$ by real numbers $M {\displaystyle M}$ and $x {\displaystyle x}$ , respectively) yields the definition of thelimit of $f(x)$ as $x {\displaystyle x}$ increases without bound, notated ${\textstyle \lim _{x\to \infty }f(x)}$ . Reversing the inequality $x\geq M$ to $x\leq M$ gives the corresponding definition of the limit of $f(x)$ as $x {\displaystyle x}$ decreaseswithout bound, ${\textstyle \lim _{x\to -\infty }f(x)}$ .

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 aCauchy sequence if, for any $\varepsilon >0$ , there exists a natural number $N {\displaystyle N}$ 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, $(\mathbb {R} ,|\cdot |)$ , is acomplete 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

[edit]

Main article:Uniform convergence

In addition to sequences of numbers, one may also speak ofsequences of functionson $E\subset \mathbb {R}$ , that is, infinite, ordered families of functions $f_{n}:E\to \mathbb {R}$ , denoted $(f_{n})_{n=1}^{\infty }$ , and their convergence properties. However, in the case of sequences of functions, there are two kinds of convergence, known aspointwise convergence anduniform convergence, that need to be distinguished.

Roughly speaking, pointwise convergence of functions $f_{n}$ to a limiting function $f:E\to \mathbb {R}$ , 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 $f {\displaystyle f}$ forevery value of $x\in E$ , whenever $n\geq N$ , for some integer $N {\displaystyle N}$ . For a family of functions to uniformly converge, sometimes denoted $f_{n}\rightrightarrows f$ , such a value of $N {\displaystyle N}$ must exist for any $\varepsilon >0$ given, no matter how small. Intuitively, we can visualize this situation by imagining that, for a large enough $N {\displaystyle N}$ , the functions $f_{N},f_{N+1},f_{N+2},\ldots$ are all confined within a 'tube' of width $2\varepsilon$ about $f {\displaystyle f}$ (that is, between $f-\varepsilon$ and $f+\varepsilon$ )for every value in their domain $E {\displaystyle E}$ .

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 (seebelow) 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

[edit]

Main article:Compactness

Compactness is a concept fromgeneral 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 beingclosed andbounded. (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, aclosed set contains all of itsboundary points, while a set isbounded if there exists a real number such that the distance between any two points of the set is less than that number. In $\mathbb {R}$ , 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 $\{1/n:n\in \mathbb {N} \}\cup \{0}\$ is a compact set; theCantor ternary set ${\mathcal {C}}\subset [0,1]$ is another example of a compact set. On the other hand, the set $\{1/n:n\in \mathbb {N} \}$ is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set. The set $[0,\infty )$ is also not compact because it is closed but not bounded.

For subsets of the real numbers, there are several equivalent definitions of compactness.

Definition. A set $E\subset \mathbb {R}$ is compact if it is closed and bounded.

This definition also holds for Euclidean space of any finite dimension, $\mathbb {R} ^{n}$ , but it is not valid for metric spaces in general. The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as theHeine-Borel theorem.

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

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

This particular property is known assubsequential compactness. In $\mathbb {R}$ , 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 ofopen covers andsubcovers, which is applicable to topological spaces (and thus to metric spaces and $\mathbb {R}$ as special cases). In brief, a collection of open sets $U_{\alpha }$ is said to be anopen cover of set $X {\displaystyle X}$ if the union of these sets is a superset of $X {\displaystyle X}$ . This open cover is said to have afinite subcover if a finite subcollection of the $U_{\alpha }$ could be found that also covers $X {\displaystyle X}$ .

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

[edit]

Main article:Continuous function

Afunction from the set ofreal numbers to the real numbers can be represented by agraph in theCartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbrokencurve with no "holes" or "jumps".

There are several ways to make this intuition mathematically rigorous. Several definitions of varying levels of generality can be given. In cases where two or more definitions are applicable, they are readily shown to beequivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the first definition given below, $f:I\to \mathbb {R}$ is a function defined on a non-degenerate interval $I {\displaystyle I}$ of the set of real numbers as its domain. Some possibilities include $I=\mathbb {R}$ , the whole set of real numbers, anopen interval $I=(a,b)=\{x\in \mathbb {R} \mid a<x<b\},$ or aclosed interval $I=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\}.$ Here, $a {\displaystyle a}$ and $b {\displaystyle b}$ are distinct real numbers, and we exclude the case of $I {\displaystyle I}$ being empty or consisting of only one point, in particular.

Definition. If $I\subset \mathbb {R}$ is a non-degenerate interval, we say that $f:I\to \mathbb {R}$ iscontinuous at $p\in I$ if ${\textstyle \lim _{x\to p}f(x)=f(p)}$ . We say that $f {\displaystyle f}$ is acontinuous map if $f {\displaystyle f}$ is continuous at every $p\in I$ .

In contrast to the requirements for $f {\displaystyle f}$ to have a limit at a point $p {\displaystyle p}$ , which do not constrain the behavior of $f {\displaystyle f}$ at $p {\displaystyle p}$ itself, the following two conditions, in addition to the existence of ${\textstyle \lim _{x\to p}f(x)}$ , must also hold in order for $f {\displaystyle f}$ to be continuous at $p {\displaystyle p}$ :(i) $f {\displaystyle f}$ must be defined at $p {\displaystyle p}$ , i.e., $p {\displaystyle p}$ is in the domain of $f {\displaystyle f}$ ;and(ii) $f(x)\to f(p)$ as $x\to p$ . The definition above actually applies to any domain $E {\displaystyle E}$ that does not contain anisolated point, or equivalently, $E {\displaystyle E}$ where every $p\in E$ is alimit point of $E {\displaystyle E}$ . A more general definition applying to $f:X\to \mathbb {R}$ with a general domain $X\subset \mathbb {R}$ is the following:

Definition. If $X {\displaystyle X}$ is an arbitrary subset of $\mathbb {R}$ , we say that $f:X\to \mathbb {R}$ iscontinuous at $p\in X$ if, for any $\varepsilon >0$ , there exists $\delta >0$ such that for all $x\in X$ , $|x-p|<\delta$ implies that $|f(x)-f(p)|<\varepsilon$ . We say that $f {\displaystyle f}$ is acontinuous map if $f {\displaystyle f}$ is continuous at every $p\in X$ .

A consequence of this definition is that $f {\displaystyle f}$ istrivially continuous at any isolated point $p\in X$ . This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps betweentopological spaces (which includesmetric spaces and $\mathbb {R}$ in particular as special cases). This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.

Definition. If $X {\displaystyle X}$ and $Y {\displaystyle Y}$ are topological spaces, we say that $f:X\to Y$ iscontinuous at $p\in X$ if $f^{-1}(V)$ is aneighborhood of $p {\displaystyle p}$ in $X {\displaystyle X}$ for every neighborhood $V {\displaystyle V}$ of $f(p)$ in $Y {\displaystyle Y}$ . We say that $f {\displaystyle f}$ is acontinuous map if $f^{-1}(U)$ is open in $X {\displaystyle X}$ for every $U {\displaystyle U}$ open in $Y {\displaystyle Y}$ .

(Here, $f^{-1}(S)$ refers to thepreimage of $S\subset Y$ under $f {\displaystyle f}$ .)

Uniform continuity

[edit]

Main article:Uniform continuity

Definition. If $X {\displaystyle X}$ is a subset of thereal numbers, we say a function $f:X\to \mathbb {R}$ isuniformly continuouson $X {\displaystyle X}$ if, for any $\varepsilon >0$ , there exists a $\delta >0$ such that for all $x,y\in X$ , $|x-y|<\delta$ implies that $|f(x)-f(y)|<\varepsilon$ .

Explicitly, when a function is uniformly continuous on $X {\displaystyle X}$ , the choice of $\delta$ needed to fulfill the definition must work forall of $X {\displaystyle X}$ for a given $\varepsilon$ . In contrast, when a function is continuous at every point $p\in X$ (or said to be continuous on $X {\displaystyle X}$ ), the choice of $\delta$ may depend on both $\varepsilon$ and $p {\displaystyle p}$ . In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point $p {\displaystyle p}$ is meaningless.

On a compact set, it is easily shown that all continuous functions are uniformly continuous. If $E {\displaystyle E}$ is a bounded noncompact subset of $\mathbb {R}$ , then there exists $f:E\to \mathbb {R}$ that is continuous but not uniformly continuous. As a simple example, consider $f:(0,1)\to \mathbb {R}$ defined by $f(x)=1/x$ . By choosing points close to 0, we can always make $|f(x)-f(y)|>\varepsilon$ for any single choice of $\delta >0$ , for a given $\varepsilon >0$ .

Absolute continuity

[edit]

Main article:Absolute continuity

Definition. Let $I\subset \mathbb {R}$ be aninterval on thereal line. A function $f:I\to \mathbb {R}$ is said to beabsolutely continuouson $I {\displaystyle I}$ if for every positive number $\varepsilon$ , there is a positive number $\delta$ such that whenever a finite sequence ofpairwise disjoint sub-intervals $(x_{1},y_{1}),(x_{2},y_{2}),\ldots ,(x_{n},y_{n})$ of $I {\displaystyle I}$ satisfies^[6]

\sum _{k=1}^{n}(y_{k}-x_{k})<\delta

then

\sum _{k=1}^{n}|f(y_{k})-f(x_{k})|<\varepsilon .

Absolutely continuous functions are continuous: consider the casen = 1 in this definition. The collection of all absolutely continuous functions onI is denoted AC(I). Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.

Differentiation

[edit]

Main articles:Derivative andDifferential calculus

The notion of thederivative of a function ordifferentiability originates from the concept of approximating a function near a given point using the "best" linear approximation. This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point $a {\displaystyle a}$ , and the slope of the line is the derivative of the function at $a {\displaystyle a}$ .

A function $f:\mathbb {R} \to \mathbb {R}$ isdifferentiable at $a {\displaystyle a}$ if thelimit

f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}

exists. This limit is known as thederivative of $f {\displaystyle f}$ at $a {\displaystyle a}$ , and the function $f^{'} {\displaystyle f'}$ , possibly defined on only a subset of $\mathbb {R}$ , is thederivative (orderivative function)of $f {\displaystyle f}$ . If the derivative exists everywhere, the function is said to bedifferentiable.

As a simple consequence of the definition, $f {\displaystyle f}$ is continuous at $a {\displaystyle a}$ if it is differentiable there. Differentiability is therefore a stronger regularity condition (condition describing the "smoothness" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (seeWeierstrass's nowhere differentiable continuous function). It is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.

One can classify functions by theirdifferentiability class. The class $C^{0}$ (sometimes $C^{0}([a,b])$ to indicate the interval of applicability) consists of all continuous functions. The class $C^{1}$ consists of alldifferentiable functions whose derivative is continuous; such functions are calledcontinuously differentiable. Thus, a $C^{1}$ function is exactly a function whose derivative exists and is of class $C^{0}$ . In general, the classes $C^{k}$ can be definedrecursively by declaring $C^{0}$ to be the set of all continuous functions and declaring $C^{k}$ for any positive integer $k {\displaystyle k}$ to be the set of all differentiable functions whose derivative is in $C^{k-1}$ . In particular, $C^{k}$ is contained in $C^{k-1}$ for every $k {\displaystyle k}$ , and there are examples to show that this containment is strict. Class $C^{\infty }$ is the intersection of the sets $C^{k}$ as $k {\displaystyle k}$ varies over the non-negative integers, and the members of this class are known as thesmooth functions. Class $C^{\omega }$ consists of allanalytic functions, and is strictly contained in $C^{\infty }$ (seebump function for a smooth function that is not analytic).

Series

[edit]

Main article:Series (mathematics)

A series formalizes the imprecise notion of taking the sum of an endless sequence of numbers. The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers. The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms. Instead, the finite sum of the first $n {\displaystyle n}$ terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as $n {\displaystyle n}$ grows without bound. The series is assigned the value of this limit, if it exists.

Given an (infinite)sequence $(a_{n})$ , we can define an associatedseries as the formal mathematical object ${\textstyle a_{1}+a_{2}+a_{3}+\cdots =\sum _{n=1}^{\infty }a_{n}}$ , sometimes simply written as ${\textstyle \sum a_{n}}$ . Thepartial sums of a series ${\textstyle \sum a_{n}}$ are the numbers ${\textstyle s_{n}=\sum _{j=1}^{n}a_{j}}$ . A series ${\textstyle \sum a_{n}}$ is said to beconvergent if the sequence consisting of its partial sums, $(s_{n})$ , is convergent; otherwise it isdivergent. Thesum of a convergent series is defined as the number ${\textstyle s=\lim _{n\to \infty }s_{n}}$ .

The word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms. For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on theRiemann rearrangement theorem for further discussion).

An example of a convergent series is ageometric series which forms the basis of one of Zeno's famousparadoxes:

\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots =1.

In contrast, theharmonic series has been known since the Middle Ages to be a divergent series:

\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty .

(Here, " $=\infty$ " is merely a notational convention to indicate that the partial sums of the series grow without bound.)

A series ${\textstyle \sum a_{n}}$ is said toconverge absolutely if ${\textstyle \sum |a_{n}|}$ is convergent. A convergent series ${\textstyle \sum a_{n}}$ for which ${\textstyle \sum |a_{n}|}$ diverges is said toconvergenon-absolutely.^[7] It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an example of a series that converges non-absolutely is

\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots =\ln 2.

Taylor series

[edit]

Main article:Taylor series

The Taylor series of areal orcomplex-valued functionƒ(x) that isinfinitely differentiable at areal orcomplex numbera is thepower series

f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f^{(3)}(a)}{3!}}(x-a)^{3}+\cdots .

which can be written in the more compactsigma notation as

\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}\,(x-a)^{n}

wheren! denotes thefactorial ofn andƒ⁽ⁿ⁾(a) denotes thenthderivative ofƒ evaluated at the pointa. The derivative of order zeroƒ is defined to beƒ itself and(x −a)⁰ and 0! are both defined to be 1. In the case thata = 0, the series is also called a Maclaurin series.

A Taylor series off about pointa may diverge, converge at only the pointa, converge for allx such that $|x-a|<R$ (the largest suchR for which convergence is guaranteed is called theradius of convergence), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at that point. If the Taylor series at a point has a nonzeroradius of convergence, and sums to the function in thedisc of convergence, then the function isanalytic. The analytic functions have many fundamental properties. In particular, an analytic function of a real variable extends naturally to a function of a complex variable. It is in this way that theexponential function, thelogarithm, thetrigonometric functions and theirinverses are extended to functions of a complex variable.

Fourier series

[edit]

Main article:Fourier series

The first four partial sums of theFourier series for asquare wave. Fourier series are an important tool in real analysis.

Fourier series decomposesperiodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namelysines and cosines (orcomplex exponentials). The study of Fourier series typically occurs and is handled within the branchmathematics >mathematical analysis >Fourier analysis.

Integration

[edit]

Integration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface. The basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as themethod of exhaustion. Generally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed. By considering approximations consisting of a larger and larger ("infinite") number of smaller and smaller ("infinitesimal") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.

The spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices ("refinements") are considered. Though the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind. Compared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a "measure" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist "non-measurable" subsets for which an area cannot be assigned.

Riemann integration

[edit]

Main article:Riemann integral

The Riemann integral is defined in terms ofRiemann sums of functions with respect to tagged partitions of an interval. Let $[a,b]$ be aclosed interval of the real line; then atagged partition ${\cal {P}}$ of $[a,b]$ is a finite sequence

a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\,\!

This partitions the interval $[a,b]$ into $n {\displaystyle n}$ sub-intervals $[x_{i-1},x_{i}]$ indexed by $i=1,\ldots ,n$ , each of which is "tagged" with a distinguished point $t_{i}\in [x_{i-1},x_{i}]$ . For a function $f {\displaystyle f}$ bounded on $[a,b]$ , we define theRiemann sum of $f {\displaystyle f}$ with respect to tagged partition ${\cal {P}}$ as

\sum _{i=1}^{n}f(t_{i})\Delta _{i},

where $\Delta _{i}=x_{i}-x_{i-1}$ is the width of sub-interval $i {\displaystyle i}$ . Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Themesh of such a tagged partition is the width of the largest sub-interval formed by the partition, ${\textstyle \|\Delta _{i}\|=\max _{i=1,\ldots ,n}\Delta _{i}}$ . We say that theRiemann integral of $f {\displaystyle f}$ on $[a,b]$ is $S {\displaystyle S}$ if for any $\varepsilon >0$ there exists $\delta >0$ such that, for any tagged partition ${\cal {P}}$ with mesh $\|\Delta _{i}\|<\delta$ , we have

\left|S-\sum _{i=1}^{n}f(t_{i})\Delta _{i}\right|<\varepsilon .

This is sometimes denoted ${\textstyle {\mathcal {R}}\int _{a}^{b}f=S}$ . When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower)Darboux sum. A function isDarboux integrable if the upper and lowerDarboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal. In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.

Thefundamental theorem of calculus asserts that integration and differentiation are inverse operations in a certain sense.

Lebesgue integration and measure

[edit]

Main article:Lebesgue integral

Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends thedomains on which these functions can be defined. The concept of ameasure, an abstraction of length, area, or volume, is central to Lebesgue integralprobability theory.

Distributions

[edit]

Main article:Distribution (mathematics)

Distributions (orgeneralized functions) are objects that generalizefunctions. Distributions make it possible todifferentiate functions whose derivatives do not exist in the classical sense. In particular, anylocally integrable function has a distributional derivative.

Relation to complex analysis

[edit]

Real analysis is an area ofanalysis that studies concepts such as sequences and their limits, continuity,differentiation,integration and sequences of functions. By definition, real analysis focuses on thereal numbers, often including positive and negativeinfinity to form theextended real line. Real analysis is closely related tocomplex analysis, which studies broadly the same properties ofcomplex numbers. In complex analysis, it is natural to definedifferentiation viaholomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility aspower series, and satisfying theCauchy integral formula.

In real analysis, it is usually more natural to considerdifferentiable,smooth, orharmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as thefundamental theorem of algebra are simpler when expressed in terms of complex numbers.

Techniques from thetheory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals byresidue calculus.

Important results

[edit]

Important results include theBolzano–Weierstrass andHeine–Borel theorems, theintermediate value theorem andmean value theorem,Taylor's theorem, thefundamental theorem of calculus, theArzelà-Ascoli theorem, theStone-Weierstrass theorem,Fatou's lemma, and themonotone convergence anddominated convergence theorems.

Generalizations and related areas of mathematics

[edit]

Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis tometric spaces andtopological spaces connects real analysis to the field ofgeneral topology, while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts ofBanach spaces andHilbert spaces and, more generally tofunctional analysis.Georg Cantor's investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth tonaive set theory. The study of issues ofconvergence for sequences of functions eventually gave rise toFourier analysis as a subdiscipline of mathematical analysis. Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept ofholomorphic functions and the inception ofcomplex analysis as another distinct subdiscipline of analysis. On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstractmeasure spaces, a fundamental concept inmeasure theory. Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study ofvector calculus, whose further generalization and formalization played an important role in the evolution of the concepts ofdifferential forms andsmooth (differentiable) manifolds indifferential geometry and other closely related areas ofgeometry andtopology.

References

[edit]

^Tao, Terence (2003)."Lecture notes for MATH 131AH"(PDF).Course Website for MATH 131AH, Department of Mathematics, UCLA.
^"Sequences intro".khanacademy.org.
^Gaughan, Edward (2009). "1.1 Sequences and Convergence".Introduction to Analysis. AMS (2009).ISBN 978-0-8218-4787-9.
^Some authors (e.g., Rudin 1976) use braces instead and write $\{a_{n}\}$ . However, this notation conflicts with the usual notation for aset, which, in contrast to a sequence, disregards the order and the multiplicity of its elements.
^Stewart, James (2008).Calculus: Early Transcendentals (6th ed.).Brooks/Cole.ISBN 978-0-495-01166-8.
^Royden 1988, Sect. 5.4, page 108;Nielsen 1997, Definition 15.6 on page 251;Athreya & Lahiri 2006, Definitions 4.4.1, 4.4.2 on pages 128,129. The intervalI is assumed to be bounded and closed in the former two books but not the latter book.
^The termunconditional convergence refers to series whose sum does not depend on the order of the terms (i.e., any rearrangement gives the same sum). Convergence is termedconditional otherwise. For series in $\mathbb {R} ^{n}$ , it can be shown that absolute convergence and unconditional convergence are equivalent. Hence, the term "conditional convergence" is often used to mean non-absolute convergence. However, in the general setting of Banach spaces, the terms do not coincide, and there are unconditionally convergent series that do not converge absolutely.

Sources

[edit]

Athreya, Krishna B.; Lahiri, Soumendra N. (2006),Measure theory and probability theory, Springer,ISBN 0-387-32903-X
Nielsen, Ole A. (1997),An introduction to integration and measure theory, Wiley-Interscience,ISBN 0-471-59518-7
Royden, H.L. (1988),Real Analysis (third ed.), Collier Macmillan,ISBN 0-02-404151-3

Bibliography

[edit]

Abbott, Stephen (2001).Understanding Analysis. Undergraduate Texts in Mathematics. New York: Springer-Verlag.ISBN 0-387-95060-5.
Aliprantis, Charalambos D.; Burkinshaw, Owen (1998).Principles of real analysis (3rd ed.). Academic.ISBN 0-12-050257-7.
Bartle, Robert G.; Sherbert, Donald R. (2011).Introduction to Real Analysis (4th ed.). New York: John Wiley and Sons.ISBN 978-0-471-43331-6.
Bressoud, David (2007).A Radical Approach to Real Analysis. MAA.ISBN 978-0-88385-747-2.
Browder, Andrew (1996).Mathematical Analysis: An Introduction.Undergraduate Texts in Mathematics. New York: Springer-Verlag.ISBN 0-387-94614-4.
Carothers, Neal L. (2000).Real Analysis. Cambridge: Cambridge University Press.ISBN 978-0521497565.
Dangello, Frank; Seyfried, Michael (1999).Introductory Real Analysis. Brooks Cole.ISBN 978-0-395-95933-6.
Kolmogorov, A. N.;Fomin, S. V. (1975).Introductory Real Analysis. Translated by Richard A. Silverman. Dover Publications.ISBN 0486612260. Retrieved2 April 2013.
Rudin, Walter (1976).Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill.ISBN 978-0-07-054235-8.
Rudin, Walter (1987).Real and Complex Analysis (3rd ed.). New York: McGraw-Hill.ISBN 978-0-07-054234-1.
Spivak, Michael (1994).Calculus (3rd ed.).Houston, Texas: Publish or Perish, Inc.ISBN 091409890X.

External links

[edit]

How We Got From There to Here: A Story of Real Analysis by Robert Rogers and Eugene Boman
A First Course in Analysis by Donald Yau
Analysis WebNotes by John Lindsay Orr
Interactive Real Analysis by Bert G. Wachsmuth
A First Analysis Course by John O'Connor
Mathematical Analysis I by Elias Zakon
Mathematical Analysis II by Elias Zakon
Trench, William F. (2003).Introduction to Real Analysis(PDF).Prentice Hall.ISBN 978-0-13-045786-8.
Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
Basic Analysis: Introduction to Real Analysis by Jiri Lebl
Topics in Real Analysis byGerald Teschl, University of Vienna.

v t e Major topics inmathematical analysis
Calculus:Integration Differentiation Differential equations ordinary partial stochastic Fundamental theorem of calculus Calculus of variations Vector calculus Tensor calculus Matrix calculus Lists of integrals Table of derivatives
Real analysis Complex analysis Hypercomplex analysis (quaternionic analysis) Functional analysis Fourier analysis Least-squares spectral analysis Harmonic analysis P-adic analysis (P-adic numbers) Measure theory Representation theory Functions Continuous function Special functions Limit Series Infinity
Mathematics portal