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Real analysis

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Real analysis

The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

Scope

Construction of the real numbers

There are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in more detail in the main article.

Order properties of the real numbers

The real numbers have several important lattice-theoretic properties that are absent in the complex numbers. Most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property. These order-theoretic properties lead to a number of important 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.

Sequences

A sequence is usually defined as a function whose domain is a countable totally ordered set, although in many disciplines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n) : NR. We might identify an = f(n)   for all n or just write an : NR.

Limits

A limit is the value that a function or sequence "approaches" as the input or index approaches some value.[2] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

Continuity

A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".

There are several ways to make this intuition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the definitions below,

f\colon I \rightarrow \mathbf R.

is a function defined on a subset I of the set R of real numbers. This subset I is referred to as the domain of f. Some possible choices include I=R, the whole set of real numbers, an open interval

I = (a, b) = \{x \in \mathbf R \,|\, a < x < b \},

or a closed interval

I = [a, b] = \{x \in \mathbf R \,|\, a \leq x \leq b \}.

Here, a and b are real numbers.

Uniform continuity

If X and Y are subsets of the real numbers, a function f : X → Y is called uniformly continuous if for all ε > 0 there exists a δ > 0 such that for all xy ∈ X, |x − y| < δ implies |f(x) − f(y)| < ε.

The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.

Absolute continuity

Let I be an interval in the real line R. A function f:I \to R is absolutely continuous on I if for every positive number \epsilon, there is a positive number \delta such that whenever a finite sequence of pairwise disjoint sub-intervals (x_k, y_k) of I satisfies[3]

\sum_{k} \left| y_k - x_k \right| < \delta

then

\displaystyle \sum_{k} | f(y_k) - f(x_k) | < \epsilon.

The collection of all absolutely continuous functions on I is denoted AC(I).

The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:[4]

(1) f is absolutely continuous;
(2) f has a derivative f ′ almost everywhere, the derivative is Lebesgue integrable, and
f(x) = f(a) + \int_a^x f'(t) \, dt
for all x on [a,b];
(3) there exists a Lebesgue integrable function g on [a,b] such that
f(x) = f(a) + \int_a^x g(t) \, dt
for all x on [a,b].

If these equivalent conditions are satisfied then necessarily g = f ′ almost everywhere.

Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue.[5]

Series

Given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy and its mathematical representation:

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

The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm.

Taylor series

The Taylor series of a real or complex-valued function ƒ(x) that is infinitely differentiable at a real or complex number a is the power 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 compact sigma notation as

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

where n! denotes the factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a. The derivative of order zero ƒ is defined to be ƒ itself and (xa)0 and 0! are both defined to be 1. In the case that a = 0, the series is also called a Maclaurin series.

Fourier series

A Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

Differentiation

Formally, the derivative of the function f at a is the limit

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

If the derivative exists everywhere, the function is differentiable. One can take higher derivatives as well, by iterating this process.

One can classify functions by their differentiability class. The class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k, and there are examples to show that this containment is strict. C is the intersection of the sets Ck as k varies over the non-negative integers. Cω is strictly contained in C.

Integration

Riemann integration

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence

a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!

This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as

\sum_{i=1}^{n} f(t_i) \Delta_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. Let Δi = xixi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1...n Δi. The Riemann integral of a function f over the interval [a,b] is equal to S if:

For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have
\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \varepsilon.

When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.

Lebesgue integration

Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined.

Distributions

Distributions (or generalized functions) are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

Relation to complex analysis

Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line. Real analysis is closely related to complex analysis, which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressability as power series, and satisfying the Cauchy integral formula.

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

Techniques from the theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by residue calculus.

Important results

Important results include the Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem, the fundamental theorem of calculus, and the monotone convergence theorem.

Various ideas from real analysis can be generalized from real space to general metric spaces, as well as to measure spaces, Banach spaces, and Hilbert spaces.

See also

References

  1. ^
  2. ^
  3. ^ 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 interval I is assumed to be bounded and closed in the former two books but not the latter book.
  4. ^ Nielsen 1997, Theorem 20.8 on page 354; also Royden 1988, Sect. 5.4, page 110 and Athreya & Lahiri 2006, Theorems 4.4.1, 4.4.2 on pages 129,130.
  5. ^ Athreya & Lahiri 2006, before Theorem 4.4.1 on page 129.

Bibliography

External links

  • 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
  • Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
  • Basic Analysis: Introduction to Real Analysis by Jiri Lebl
  • Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna.
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