# List of real analysis topics

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List of real analysis topics

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

## General topics

### Sequences and Series

#### Summation Methods

• Convolution
• 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.

### Integrals

• 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

## 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.

## Applied mathematical tools

### Inequalities

See list of inequalities

### 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.

## 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 limit-related structures and the linear operators acting upon these spaces

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