Real analysis 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. In other words, a sequence is a map f(n) : N → R. We might identify
an = f(n) for all n or just write an : N → R.
= Limits= A limit is the value that a function or
sequence "approaches" as the input or index approaches some value. Limits are
essential to calculus 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, 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
or a closed interval 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 x, y ∈ X, |x − y| 0 there exists δ > 0
such that, for any tagged partition [a,b] with mesh less than δ, we have
When the chosen tags give the maximum value of each interval, the Riemann sum
becomes an upper 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 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 List of real analysis topics
Time-scale calculus – a unification of real analysis with calculus of finite
differences Real multivariable function
Real coordinate space Complex analysis
References Bibliography
Aliprantis, Charalambos D.; Burkinshaw, Owen. Principles of real analysis.
Academic. ISBN 0-12-050257-7. Browder, Andrew. Mathematical Analysis:
An Introduction. Undergraduate Texts in Mathematics. New York: Springer-Verlag.
ISBN 0-387-94614-4. Bartle, Robert G.; Sherbert, Donald R..
Introduction to Real Analysis. New York: John Wiley and Sons. ISBN 0-471-32148-6.
Abbott, Stephen. Understanding Analysis. Undergradutate Texts in Mathematics. New
York: Springer-Verlag. ISBN 0-387-95060-5.
Rudin, Walter. Principles of Mathematical Analysis. Walter Rudin
Student Series in Advanced Mathematics. McGraw–Hill. ISBN 978-0-07-054235-8.
Dangello, Frank; Seyfried, Michael. Introductory Real Analysis. Brooks Cole.
ISBN 978-0-395-95933-6. Bressoud, David. A Radical Approach to
Real Analysis. MAA. ISBN 0-88385-747-2. Kolmogorov, A. N.; Fomin, S. V..
Introductory Real Analysis. Translated by Richard A. Silverman. Dover
Publications. ISBN 0486612260. Retrieved 2 April 2013.
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 Trench, William F.. Introduction to Real
Analysis. 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 and Functional Analysis by Gerald Teschl, University of Vienna.