List of real analysis topics

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

Contents 1 General topics 1.1 Limits 1.2 Sequences and Series 1.2.1 Summation Methods 1.2.2 More advanced topics 1.3 Convergence 1.3.1 Convergence tests 1.4 Functions 1.4.1 Continuity 1.4.2 Distributions 1.4.3 Variation 1.5 Derivatives 1.5.1 Differentiation rules 1.5.2 Differentiation in Geometry and Topology 1.6 Integrals 1.6.1 Integration and Measure theory 2 Fundamental theorems 3 Foundational topics 3.1 Numbers 3.1.1 Real numbers 3.1.2 Specific Numbers 3.2 Sets 3.3 Maps 4 Applied mathematical tools 4.1 Infinite expressions 4.2 Inequalities 4.3 Means 4.4 Orthogonal polynomials 4.5 Spaces 4.6 Measures 4.7 Field of sets 5 Historical figures 6 Related fields of analysis

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
  • Taylor series - a series of the form
    • 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 - algerbraic 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
• Comparison test
• Root test
• Alternating series test
• Cauchy condensation test
• Abel's test
• Dirichlet's test
• Stolz–Cesàro theorem - is a criterion for proving the convergence of a sequence

Functions

• Function of a real variable
• 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 [0, ∞) 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 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

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'Hopital'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 Rⁿ has a convergent subsequence.

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
• Triangle 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 (1736-1813)
• Jean Baptiste Joseph Fourier (1768-1830)
• Bernard Bolzano (1781-1848)
• Augustin Cauchy (1789-1857)
• Niels Henrik Abel (1802-1829)
• Johann 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)
• Waclaw Sierpinski (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