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 $f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots$
- Taylor series – a series of the form $f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f^{(3)}(a)}{3!}}(x-a)^{3}+\cdots .$
  - 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

Convergence

Distributions

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

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 anitderivatives
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

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 $\mathbb {R} ^{n}$ has a convergent subsequence.
Extreme value theorem - states that if a function $f$ is continuous in the closed and bounded interval $[a,b]$ , then it must attain a maximum and a minimum, each at least once.

Foundational topics

Numbers

Real numbers

Specific numbers

Sets

Maps

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

Applied mathematical tools

Infinite expressions

Inequalities

See list of inequalities

Means

Orthogonal polynomials

Classical orthogonal 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 (1736–1813)
Joseph Fourier (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

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