List of real analysis topics

This is a list of articles that are considered real analysis topics.

General topics

Limits

*Limit of a sequence
** Subsequential limit – the limit of some subsequence
*Limit of a function (''see List of limits for a list of limits of common functions)
**One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below
**Squeeze theorem – confirms the limit of a function via comparison with two other functions
**Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions

Sequences and series

(''see also list of mathematical series'')

*Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant
**Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
*Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
* Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression
*Finite sequence – ''see sequence''
*Infinite sequence – ''see sequence''
*Divergent sequence – ''see limit of a sequence or divergent series''
*Convergent sequence – ''see limit of a sequence or convergent series''
**Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses
*Convergent series – a series whose sequence of partial sums converges
*Divergent series – a series whose sequence of partial sums diverges
*Power series – a series of the form $f(x) = \sum_^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots$
**Taylor series – a series of the form $f(a)+\frac (x-a)+ \frac (x-a)^2+\frac(x-a)^3+ \cdots.$
***Maclaurin series – ''see Taylor series''
****Binomial series – the Maclaurin series of the function ''f'' given by ''f''(''x'') ''='' (1 + ''x'') ^''α''
*Telescoping series
*Alternating series
*Geometric series
**Divergent geometric series
* Harmonic series
*Fourier series
* Lambert series

Summation methods

*Cesàro summation
*Euler summation
* Lambert summation
*Borel summation
*Summation by parts – transforms the summation of products of into other summations
* Cesàro mean
*Abel's summation formula

More advanced topics

*Convolution
**Cauchy product –is the discrete convolution of two sequences
*Farey sequence – the sequence of completely reduced fractions between 0 and 1
*Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
*Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 0⁰, 0/0, 1^∞, ∞ − ∞, ∞/∞, 0 × ∞, and ∞⁰.

Convergence

*Pointwise convergence, Uniform convergence
*Absolute convergence, Conditional convergence
* Normal convergence
*Radius of convergence

Convergence tests

* Integral test for convergence
*Cauchy's convergence test
*Ratio test
*Direct comparison test
* Limit comparison test
*Root test
*Alternating series test
*Dirichlet's test
*Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence

Functions

*Function of a real variable
*Real multivariable function
*Continuous function
**Nowhere continuous function
**Weierstrass function
*Smooth function
**Analytic function
***Quasi-analytic function
**Non-analytic smooth function
** Flat function
**Bump function
*Differentiable function
*Integrable function
**Square-integrable function, p-integrable function
*Monotonic function
** Bernstein's theorem on monotone functions – states that any real-valued function on the half-line , ∞) that is totally monotone is a mixture of exponential functions
*Inverse function
*Convex function, Concave function
*Singular function
*Harmonic function
**Weakly harmonic function
**Proper convex function
*Rational function
*Orthogonal function
*Implicit and explicit functions
** Implicit function theorem – allows relations to be converted to functions
*Measurable function
*Baire one star function
*Symmetric function
*Domain
*Codomain
**Image
*Support
*Differential of a function

Continuity

*Uniform continuity
**Modulus of continuity
*Lipschitz continuity
*Semi-continuity
*Equicontinuous
*Absolute continuity
*Hölder condition – condition for Hölder continuity

Distributions

*Dirac delta function
*Heaviside step function
*Hilbert transform
*Green's function

Variation

* Bounded variation
*Total variation

Derivatives

*Second derivative
**Inflection point – found using second derivatives
*Directional derivative, Total derivative, Partial derivative

Differentiation rules

* Linearity of differentiation
*Product rule
*Quotient rule
*Chain rule
*Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function

Differentiation in geometry and topology

''see also List of differential geometry topics''

*Differentiable manifold
*Differentiable structure
* Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

Integrals

''(see also Lists of integrals)''

*Antiderivative
**Fundamental theorem of calculus – a theorem of antiderivatives
*Multiple integral
*Iterated integral
*Improper integral
**Cauchy principal value – method for assigning values to certain improper integrals
*Line integral
* Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an ''n''-dimensional convex body (''K'') does not decrease if ''K'' is translated inwards towards the origin

Integration and measure theory

''see also List of integration and measure theory topics''

*Riemann integral, Riemann sum
**Riemann–Stieltjes integral
*Darboux integral
*Lebesgue integration

Fundamental theorems

*Monotone convergence theorem – relates monotonicity with convergence
*Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
*Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
*Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc
*Taylor's theorem – gives an approximation of a $k$ times differentiable function around a given point by a $k$-th order Taylor-polynomial.
*L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms
*Abel's theorem – relates the limit of a power series to the sum of its coefficients
*Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function
*Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
* Heine–Borel theorem – sometimes used as the defining property of compactness
*Bolzano–Weierstrass theorem – states that each bounded sequence in $\mathbb^$ has a convergent subsequence
*Extreme value theorem - states that if a function $f$ is continuous in the closed and bounded interval ,b/math>, then it must attain a maximum and a minimum

Foundational topics

Numbers

Real numbers

*Construction of the real numbers
**Natural number
**Integer
**Rational number
**Irrational number
*Completeness of the real numbers
*Least-upper-bound property
*Real line
**Extended real number line
**Dedekind cut

Specific numbers

* 0
* 1
** 0.999...
*Infinity

Sets

*Open set
*Neighbourhood
*Cantor set
*Derived set (mathematics)
* Completeness
*Limit superior and limit inferior
**Supremum
**Infimum
* Interval
**Partition of an interval

Maps

*Contraction mapping
*Metric map
* Fixed point – a point of a function that maps to itself

Applied mathematical tools

Infinite expressions

*Continued fraction
*Series
*Infinite products

Inequalities

''See list of inequalities''

*Triangle inequality
*Bernoulli's inequality
*Cauchy–Schwarz inequality
* Hölder's inequality
*Minkowski inequality
*Jensen's inequality
*Chebyshev's inequality
*Inequality of arithmetic and geometric means

Means

*Generalized mean
* Pythagorean means
**Arithmetic mean
**Geometric mean
**Harmonic mean
* Geometric–harmonic mean
* Arithmetic–geometric mean
*Weighted mean
*Quasi-arithmetic mean

Orthogonal polynomials

*Classical orthogonal polynomials
**Hermite polynomials
**Laguerre polynomials
**Jacobi polynomials
** Gegenbauer polynomials
**Legendre polynomials

Spaces

*Euclidean space
*Metric space
** Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
**Complete metric space
*Topological space
**Function space
***Sequence space
*Compact space

Measures

*Lebesgue measure
*Outer measure
**Hausdorff measure
*Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

Field of sets

* Sigma-algebra

Historical figures

*Michel Rolle (1652–1719)
*Brook Taylor (1685–1731)
*Leonhard Euler (1707–1783)
*Joseph-Louis Lagrange (1768–1830)
*Bernard Bolzano (1781–1848)
*Augustin Cauchy (1789–1857)
*Niels Henrik Abel (1802–1829)
*Peter Gustav Lejeune Dirichlet (1805–1859)
*Karl Weierstrass (1815–1897)
*Eduard Heine (1821–1881)
*Pafnuty Chebyshev (1821–1894)
*Leopold Kronecker (1823–1891)
*Bernhard Riemann (1826–1866)
*Richard Dedekind (1831–1916)
*Rudolf Lipschitz (1832–1903)
*Camille Jordan (1838–1922)
*Jean Gaston Darboux (1842–1917)
*Georg Cantor (1845–1918)
*Ernesto Cesàro (1859–1906)
*Otto Hölder (1859–1937)
*Hermann Minkowski (1864–1909)
*Alfred Tauber (1866–1942)
*Felix Hausdorff (1868–1942)
*Émile Borel (1871–1956)
*Henri Lebesgue (1875–1941)
*Wacław Sierpiński (1882–1969)
*Johann Radon (1887–1956)
*Karl Menger (1902–1985)

Related fields of analysis

*Asymptotic analysis – studies a method of describing limiting behaviour
*Convex analysis – studies the properties of convex functions and convex sets
**List of convexity topics
*Harmonic analysis – studies the representation of functions or signals as superpositions of basic waves
**List of harmonic analysis topics
*Fourier analysis – studies Fourier series and Fourier transforms
**List of Fourier analysis topics
**List of Fourier-related transforms
*Complex analysis – studies the extension of real analysis to include complex numbers
*Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
*Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals.

See also

* Calculus, the classical calculus of Newton and Leibniz.
* Non-standard calculus, a rigorous application of infinitesimals, in the sense of non-standard analysis, to the classical calculus of Newton and Leibniz.

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