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

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