Glossary of real and complex analysis

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This is a glossary of concepts and results in real analysis and complex analysis in mathematics.

Contents

See also: list of real analysis topics, list of complex analysis topics and glossary of functional analysis.

A

Abel
1.   Abel sum
2.   Abel integral
analytic capacity
analytic capacity.
analytic continuation
An analytic continuation of a holomorphic function is a unique holomorphic extension of the function (on a connected open subset of ).
argument principle
argument principle
Ascoli
Ascoli's theorem says that an equicontinous bounded sequence of functions on a compact subset of has a convergent subsequence with respect to the sup norm.

B

Borel
1.  A Borel measure is a measure whose domain is the Borel σ-algebra.
2.  The Borel σ-algebra on a topological space is the smallest σ-algebra containing all open sets.
3.   Borel's lemma says that a given formal power series, there is a smooth function whose Taylor series coincides with the given series.
bounded
A subset of a metric space is bounded if there is some such that for all .
bump
A bump function is a nonzero compactly-supported smooth function, usually constructed using the exponential function.

C

Calderón
Calderón–Zygmund lemma
capacity
Capacity of a set is a notion in potential theory.
Carathéodory
Carathéodory's extension theorem
Cartan
Cartan's theorems A and B.
Cauchy
1.  The Cauchy–Riemann equations are a system of differential equations such that a function satisfying it (in the distribution sense) is a holomorphic function.
2.   Cauchy integral formula.
3.   Cauchy residue theorem.
4.   Cauchy's estimate.
5.  The Cauchy principal value is, when possible, a number assigned to a function when the function is not integrable.
6.  On a metric space, a sequence is called a Cauchy sequence if ; i.e., for each , there is an such that for all .
Cesàro
Cesàro summation is one way to compute a divergent series.
continuous
A function between metric spaces and is continuous if for any convergent sequence in , we have in .
contour
The contour integral of a measurable function over a piece-wise smooth curve is .
converge
1.  A sequence in a topological space is said to converge to a point if for each open neighborhood of , the set is finite.
2.  A sequence in a metric space is said to converge to a point if for all , there exists an such that for all , we have .
3.  A series on a normed space (e.g., ) is said to converge if the sequence of the partial sums converges.
convolution
The convolution of two functions on a convex set is given by
provided the integration converges.
Cousin
Cousin problems.
cutoff
cutoff function.

D

Dedekind
A Dedekind cut is one way to construct real numbers.
derivative
Given a map between normed spaces, the derivative of at a point x is a (unique) linear map such that .
differentiable
A map between normed space is differentiable at a point x if the derivative at x exists.
differentiation
Lebesgue's differentiation theorem says: for almost all x.
Dini
Dini's theorem.
Dirac
The Dirac delta function on is a distribution (so not exactly a function) given as
distribution
A distribution is a type of a generalized function; precisely, it is a continuous linear functional on the space of test functions.
divergent
A divergent series is a series whose partial sum does not converge. For example, is divergent.
dominated
Lebesgue's dominated convergence theorem says converges to if is a sequence of measurable functions such that converges to pointwise and for some integrable function .

E

edge
Edge-of-the-wedge theorem.
entire
An entire function is a holomorphic function whose domain is the entire complex plane.
equicontinuous
A set of maps between fixed metric spaces is said to be equicontinuous if for each , there exists a such that for all with . A map is uniformly continuous if and only if is equicontinuous.

F

Fatou
Fatou's lemma
Fourier
1.  The Fourier transform of a function on is: (provided it makes sense)
2.  The Fourier transform of a distribution is . For example, (Fourier's inversion formula).

G

Gauss
1.  The Gauss–Green formula
2.   Gaussian kernel
generalized
A generalized function is an element of some function space that contains the space of ordinary (e.g., locally integrable) functions. Examples are Schwartz's distributions and Sato's hyperfunctions.

H

Hardy-Littlewood maximal inequality
The Hardy-Littlewood maximal function of is

The Hardy-Littlewood maximal inequality states that there is some constant such that for all and all ,

Hardy space
Hardy space
Hartogs
1.   Hartogs extension theorem
2.   Hartogs's theorem on separate holomorphicity
harmonic
A function is harmonic if it satisfies the Laplace equation (in the distribution sense if the function is not twice differentiable).
Hausdorff
The Hausdorff–Young inequality says that the Fourier transformation is a well-defined bounded operator when .
Heaviside
The Heaviside function is the function H on such that and .
Hilbert space
A Hilbert space is a real or complex inner product space that is a complete metric space with the metric induced by the inner product.
holomorphic function
A function defined on an open subset of is holomorphic if it is complex differentiable. Equivalently, a function is holomorphic if it satisfies the Cauchy–Riemann equations (in the distribution sense if the function is not differentiable).

I

integrable
A measurable function is said to be integrable if .
integral
1.  The integral of the indicator function on a measurable set is the measure (volume) of the set.
2.  The integral of a measurable function is then defined by approximating the function by linear combinations of indicator functions.
isometry
An isometry between metric spaces and is a bijection that preserves the metric: for all .

L

Lebesgue differentiation theorem
The Lebesgue differentiation theorem states that for locally integrable , the equalities

and

hold for almost every . The set where they hold is called the Lebesgue set of , and points in the Lebesgue set are called Lebesgue points.

Lebesgue integral
Lebesgue integral.
Lebesgue measure
Lebesgue measure.
Lelong
Lelong number.
Levi
Levi's problem asks to show a pseudoconvex set is a domain of holomorphy.
line integral
Line integral.
Liouville
Liouville's theorem says a bounded entire function is a constant function.
Lipschitz
1.  A map between metric spaces is said to be Lipschitz continuous if .
2.  A map is locally Lipschitz continuous if it is Lipschitz continuous on each compact subset.

M

maximum
The maximum principle says that a maximum value of a harmonic function in a connected open set is attained on the boundary.
measurable function
A measurable function is a structure-preserving function between measurable spaces in the sense that the preimage of any measurable set is measurable.
measurable set
A measurable set is an element of a σ-algebra.
measurable space
A measurable space consists of a set and a σ-algebra on that set which specifies what sets are measurable.
measure
A measure is a function on a measurable space that assigns to each measurable set a number representing its measure or size. Specifically, if X is a set and Σ is a σ-algebra on X, then a set-function μ from Σ to the extended real number line is called a measure if the following conditions hold:
  • Non-negativity: For all
  • Countable additivity (or σ-additivity): For all countable collections of pairwise disjoint sets in Σ,
measure space
A measure space consists of a measurable space and a measure on that measurable space.
meromorphic
A meromorphic function is an equivalence class of functions that are locally fractions of holomorphic functions.
method of stationary phase
The method of stationary phase.
metric space
A metric space is a set X equipped with a function , called a metric, such that (1) iff , (2) for all , (3) for all .
microlocal
The notion microlocal refers to a consideration on the cotangent bundle to a space as opposed to that on the space itself. Explicitly, it amounts to considering functions on both points and momenta; not just functions on points.
Minkowski
Minkowski inequality
monotone
Monotone convergence theorem.
Morera
Morera's theorem says a function is holomorphic if the integrations of it over arbitrary closed loops are zero.
Morse
Morse function.

N

Nash
1.   Nash function.
2.   Nash–Moser theorem.
Nevanlinna theory
Nevanlinna theory concerns meromorphic functions.
net
A net is a generalization of a sequence.
normed vector space
A normed vector space, also called a normed space, is a real or complex vector space V on which a norm is defined. A norm is a map satisfying four axioms:
  1. Non-negativity: for every ,.
  2. Positive definiteness: for every , if and only if is the zero vector.
  3. Absolute homogeneity: for every scalar and ,
  4. Triangle inequality: for every and ,

O

Oka
Oka's coherence theorem says the sheaf of holomorphic functions is coherent.
open
The open mapping theorem (complex analysis)
oscillatory integral
An oscillatory integral can give a sense to a formal integral expression like

P

Paley
Paley–Wiener theorem
phase
The phase space to a configuration space (in classical mechanics) is the cotangent bundle to .
plurisubharmonic
A function on an open subset is said to be plurisubharmonic if is subharmonic for in a neighborhood of zero in and points in .
Poisson
Poisson kernel
power series
A power series is informally a polynomial of infinite degree; i.e., .
pseudoconex
A pseudoconvex set is a generalization of a convex set.

R

Radon measure
Let be a locally compact Hausdorff space and let be a positive linear functional on the space of continuous functions with compact support . Positivity means that if . There exist Borel measures on such that for all . A Radon measure on is a Borel measure that is finite on all compact sets, outer regular on all Borel sets, and inner regular on all open sets. These conditions guarantee that there exists a unique Radon measure on such that for all .
real-analytic
A real-analytic function is a function given by a convergent power series.
Rellich
Rellich's lemma tells when an inclusion of a Sobolev space to another Sobolev space is a compact operator.
Riemann
1.  The Riemann integral of a function is either the upper Riemann sum or the lower Riemann sum when the two sums agree.
2.  The Riemann zeta function is a (unique) analytic continuation of the function (it's more traditional to write for ).
3.  The Riemann hypothesis, still a conjecture, says each nontrivial zero of the Riemann zeta function has real part equal to .
4.   Riemann's existence theorem.
Runge
1.   Runge's approximation theorem.
2.   Runge domain.

S

Sato
Sato's hyperfunction, a type of a generalized function.
Schwarz
A Schwarz function is a function that is both smooth and rapid-decay.
semianalytic
The notion of semianalytic is an analog of semialgebraic.
semicontinuous
A semicontinuous function.
sequence
A sequence on a set is a map .
series
A series is informally an infinite summation process . Thus, mathematically, specifying a series is the same as specifying the sequence of the terms in the series. The difference is that, when considering a series, one is often interested in whether the sequence of partial sums converges or not and if so, to what.
σ-algebra
A σ-algebra on a set is a nonempty collection of subsets closed under complements, countable unions, and countable intersections.
Stieltjes
Stieltjes–Vitali theorem
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is any one of a number of related generalizations of the Weierstrass approximation theorem, which states that any continuous real-valued function defined on a closed interval can be uniformly approximated by polynomials. Let be a compact Hausdorff space and let have the uniform metric. One version of the Stone–Weierstrass theorem states that if is a closed subalgebra of that separates points and contains a nonzero constant function, then in fact . If a subalgebra is not closed, taking the closure and applying the previous version of the Stone–Weierstrass theorem reveals a different version of the theorem: if is a subalgebra of that separates points and contains a nonzero constant function, then is dense in .
subanalytic
subanalytic.
subharmonic
A twice continuously differentiable function is said to be subharmonic if where is the Laplacian. The subharmonicity for a more general function is defined by a limiting process.
subsequence
A subsequence of a sequence is another sequence contained in the sequence; more precisely, it is a composition where is a strictly increasing injection and is the given sequence.
support
1.  The support of a function is the closure of the set of points where the function does not vanish.
2.  The support of a distribution is the support of it in the sense in sheaf theory.

T

Tauberian
Tauberian theory is a set of results (called tauberian theorems) concerning a divergent series; they are sort of converses to abelian theorems but with some additional conditions.
Taylor
Taylor expansion
tempered
A tempered distribution is a distribution that extends to a continuous linear functional on the space of Schwarz functions.
test
A test function is a compactly-supported smooth function.

U

uniform
1.  A sequence of maps from a topological space to a normed space is said to converge uniformly to if .
2.  A map between metric spaces is said to be uniformly continuous if for each , there exist a such that for all with .

V

Vitali covering lemma
The Vitali covering lemma states that if is a collection of open balls in and

then there exists a finite number of balls such that

W

Weierstrass
1.   Weierstrass preparation theorem.
2.   Weierstrass M-test.
Weyl
1.   Weyl calculus.
2.   Weyl quantization.
Whitney
1.  The Whitney extension theorem gives a necessary and sufficient condition for a function to be extended from a closed set to a smooth function on the ambient space.
2.   Whitney stratification

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References

Further reading