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Encyclopedia > Real analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. It is a sophisticated theory of the 'numerical function' idea, and contains modern theories of generalized functions. Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the real numbers may be described informally in several different ways. ... Partial plot of a function f. ... For other uses of Calculus, see Calculus (disambiguation) Calculus is an important branch of mathematics. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ... In calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, generalized functions are objects generalizing the notion of functions. ...

The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... In abstract mathematics, naive set theory1 was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

Then the real numbers are either introduced axiomatically, or they are constructed from Cauchy sequences or Dedekind cuts of rational numbers. Initial consequences are derived, most importantly the properties of the absolute value such as the triangle inequality and Bernoulli's inequality. This article does not cite its references or sources. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x â‰¤ a implies that x is in A as well) and B is closed upwards... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics, Bernoullis inequality is an inequality that approximates exponentiations of 1 + x. ...

The concept of convergence, central to analysis, is introduced via limits of sequences. Several laws governing the limiting process can be derived, and several limits can be computed. Infinite series, which are special sequences, are also studied at this point. Power series serve to cleanly define several central functions, such as the exponential function and the trigonometric functions. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next, such as the Bolzano-Weierstrass and Heine-Borel theorems. In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... The exponential function is one of the most important functions in mathematics. ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ... In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Ã‰mile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ...

The concept of continuity may now be defined via limits. One can show that the sum, product, composition and quotient of continuous functions is continuous, excluding at points where the denominator function has value zero, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a division is called a division by zero if the divisor is zero. ... In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ... For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...

Then one can do integration (Riemann and Lebesgue) and prove the fundamental theorem of calculus, typically using the mean value theorem. In calculus, the integral of a function is an extension of the concept of a sum. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... The integral can be interpreted as the area under a curve. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated. A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity. ... In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number such that, for all x and y in M, The smallest such value of k is called the Lipschitz constant...

One can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here. In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... As the degree of the Taylor series rises, it approaches the correct function. ...

This is a list of real analysis topics. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Results from FactBites:

 Real analysis - Wikipedia, the free encyclopedia (459 words) Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next, such as the Bolzano-Weierstrass and Heine-Borel theorems.
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