List of real analysis topics

This is a list of articles that are considered real analysis topics.
See also: glossary of real and complex analysis.

General topics

Limits">Limit (mathematics)">Limits

Limit of a sequence
*Subsequential limit – the limit of some subsequence
Limit of a function
*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

[Sequence]s and series">Series (mathematics)">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
*Taylor series – a series of the form
**Maclaurin series – see Taylor series
***Binomial series – the Maclaurin series of the function f given by f ''= 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

Convergence

Distribution">distribution (mathematics)">Distributions

Variation

[Derivative]s

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

[Integral]s

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 does not decrease if K is translated inwards towards the origin

Integration and measure theory

see also List of integration and measure theory topics

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 times differentiable function around a given point by a -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 has a convergent subsequence
Extreme value theorem - states that if a function is continuous in the closed and bounded interval, then it must attain a maximum and a minimum

Foundational topics

[Number]s

[Real number]s

Specific numbers

Sets">Set (mathematics)">Sets

Maps">Map (mathematics)">Maps

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

Applied mathematical tools

Infinite expressions">Infinite expression (mathematics)">Infinite expressions

Inequalities">Inequality (mathematics)">Inequalities

See list of inequalities

[Mean]s

[Orthogonal polynomials]

Spaces">Space (mathematics)">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">Measure (mathematics)">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

Related fields of analysis">Mathematical analysis">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.