 List of real analysis topics

This is a list of articles that are considered real analysis topics.
Contents
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)
 Onesided 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 nonzero 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}
 Maclaurin series  see Taylor series
 Taylor series  a series of the form
 Telescoping series
 Alternating series
 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 realvalued 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 0^{0}, 0/0, 1^{∞}, ∞ − ∞, ∞/∞, 0 × ∞, and ∞^{0}.
Convergence
 Pointwise convergence, Uniform convergence
 Absolute convergence, Conditional convergence
 Normal 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
 Smooth function
 Differentiable function
 Integrable function
 Squareintegrable function, pintegrable function
 Monotonic function
 Bernstein's theorem on monotone functions  states that any realvalued function on the halfline [0, ∞) that is totally monotone is a mixture of exponential functions
 Inverse function
 Convex function, Concave function
 Singular function
 Harmonic 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
Continuity
 Uniform continuity
 Semicontinuity
 Equicontinuous
 Absolute continuity
 Hölder condition  condition for Hölder continuity
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
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, nonnegative function over an ndimensional 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 kth order Taylorpolynomial.
 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
 HeineBorel theorem  sometimes used as the defining property of compactness
 BolzanoWeierstrass theorem  states that each bounded sequence in R^{n} has a convergent subsequence.
Foundational topics
Numbers
Real numbers
 Construction of the real numbers
 Completeness of the real numbers
 Leastupperbound property
 Real line
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
 Triangle inequality
 Bernoulli's inequality
 CauchySchwarz 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
 Geometricharmonic mean
 Arithmeticgeometric mean
 Weighted mean
 Quasiarithmetic mean
Orthogonal polynomials
Spaces
 Euclidean space
 Metric space
 Banach fixed point theorem  guarantees the existence and uniqueness of fixed points of certain selfmaps of metric spaces, provides method to find them
 Complete metric space
 Topological space
 Compact space
Measures
 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
Historical figures
 Michel Rolle (16521719)
 Brook Taylor (16851731)
 Leonhard Euler (17071783)
 Joseph Louis Lagrange (17361813)
 Jean Baptiste Joseph Fourier (17681830)
 Bernard Bolzano (17811848)
 Augustin Cauchy (17891857)
 Niels Henrik Abel (18021829)
 Johann Peter Gustav Lejeune Dirichlet (18051859)
 Karl Weierstrass (18151897)
 Eduard Heine (18211881)
 Pafnuty Chebyshev (18211894)
 Leopold Kronecker (18231891)
 Bernhard Riemann (18261866)
 Richard Dedekind (18311916)
 Rudolf Lipschitz (18321903)
 Camille Jordan (18381922)
 Jean Gaston Darboux (18421917)
 Georg Cantor (18451918)
 Ernesto Cesàro (18591906)
 Otto Hölder (18591937)
 Hermann Minkowski (18641909)
 Alfred Tauber (18661942)
 Felix Hausdorff (18681942)
 Émile Borel (18711956)
 Henri Lebesgue (18751941)
 Waclaw Sierpinski (18821969)
 Johann Radon (18871956)
 Karl Menger (19021985)
Related fields of analysis
 Asymptotic analysis  studies a method of describing limiting behaviour
 Convex analysis  studies the properties of convex functions and convex sets
 Harmonic analysis  studies the representation of functions or signals as superpositions of basic waves
 Fourier analysis  studies Fourier series and Fourier transforms
 Complex analysis  studies the extension of real analysis to include complex numbers
 Functional analysis  studies vector spaces endowed with limitrelated structures and the linear operators acting upon these spaces
Categories: Real analysis
 Outlines
 Mathematicsrelated lists
 Limit of a sequence
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