List of real analysis topics explained

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

See also: glossary of real and complex 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

f(x) = \sum ^{inf ty} _{n =0} a _{n} \left( x - c \right) ^{n} = a _{0} + a _{1} (x - c) ^{1} + a _{2} (x - c) ^{2} + a _{3} (x - c) ^{3} + …

• • Taylor series – a series of the form

f(a) + (f'(a))/(1 !) (x - a) + (f''(a))/(2 !) (x - a) ^{2+ (f ^{(3)} (a))/(3 !)} (x - a) ^{3+} … .

• • • 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 [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 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

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

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 (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
• 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.