Glossary of real and complex analysis

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

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 $\mathbb {C}$).

argument principle

argument principle

Ascoli

Ascoli's theorem says that an equicontinous bounded sequence of functions on a compact subset of $\mathbb {R} ^{n}$ 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 $A$ of a metric space $(X,d)$ is bounded if there is some $C>0$ such that $d(a,b)<C$ for all $a,b\in A$.

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 $x_{n}$ is called a Cauchy sequence if $d(x_{n},x_{m})\to 0$; i.e., for each $\epsilon >0$, there is an $N>0$ such that $d(x_{n},x_{m})<\epsilon$ for all $n,m\geq N$.

Cesàro

Cesàro summation is one way to compute a divergent series.

continuous

A function $f:X\to Y$ between metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ is continuous if for any convergent sequence $x_{n}\to x$ in $X$, we have $f(x_{n})\to f(x)$ in $Y$.

contour

The contour integral of a measurable function $f$ over a piece-wise smooth curve $\gamma :[0,1]\to \mathbb {C}$ is $\int _{\gamma }f\,dz:=\int _{0}^{1}\gamma ^{*}(f\,dz)$.

converge

1.  A sequence $x_{n}$ in a topological space is said to converge to a point $x$ if for each open neighborhood $U$ of $x$, the set $\{n\mid x_{n}\not \in U\}$ is finite.

2.  A sequence $x_{n}$ in a metric space is said to converge to a point $x$ if for all $\epsilon >0$, there exists an $N>0$ such that for all $n>N$, we have $d(x_{n},x)<\epsilon$.

3.  A series $x_{1}+x_{2}+\cdots$ on a normed space (e.g., $\mathbb {R} ^{n}$) is said to converge if the sequence of the partial sums $s_{n}:=\sum _{1}^{n}x_{j}$ converges.

convolution

The convolution $f*g$ of two functions on a convex set is given by

$(f*g)(x)=\int f(y-x)g(y)\,dy,$

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 $f:E\to F$ between normed spaces, the derivative of $f$ at a point x is a (unique) linear map $T:E\to F$ such that $\lim _{h\to 0}\|f(x+h)-f(x)-Th\|/\|h\|=0$.

differentiable

A map between normed space is differentiable at a point x if the derivative at x exists.

differentiation

Lebesgue's differentiation theorem says: $f(x)=\lim _{r\to 0}{\frac {1}{\operatorname {vol} (B(x,r))}}\int _{B(x,r)}f\,d\mu$ for almost all x.

Dini

Dini's theorem.

Dirac

The Dirac delta function $\delta _{0}$ on $\mathbb {R} ^{n}$ is a distribution (so not exactly a function) given as $\langle \delta _{0},\varphi \rangle =\varphi (0).$

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, $\sum _{1}^{\infty }{\frac {1}{n}}$ is divergent.

dominated

Lebesgue's dominated convergence theorem says $\int f_{n}\,d\mu$ converges to $\int f\,d\mu$ if $f_{n}$ is a sequence of measurable functions such that $f_{n}$ converges to $f$ pointwise and $|f_{n}|\leq g$ for some integrable function $g$.

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 $S$ of maps between fixed metric spaces is said to be equicontinuous if for each $\epsilon >0$, there exists a $\delta >0$ such that $\sup _{f\in S}d(f(x),f(y))<\epsilon$ for all $x,y$ with $d(x,y)<\delta$. A map $f$ is uniformly continuous if and only if $\{f\}$ is equicontinuous.

F

Fatou

Fatou's lemma

Fourier

1.  The Fourier transform of a function $f$ on $\mathbb {R} ^{n}$ is: (provided it makes sense)

${\widehat {f}}(\xi )=\int f(x)e^{-2\pi ix\cdot \xi }\,dx.$

2.  The Fourier transform ${\widehat {f}}$ of a distribution $f$ is $\langle {\widehat {f}},\varphi \rangle =\langle f,{\widehat {\varphi }}\rangle$. For example, ${\widehat {\delta _{0}}}=1$ (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 $f\in L^{1}(\mathbb {R} ^{n})$ is

$Hf(x):=\sup _{r>0}{\frac {1}{m(B_{r}(x))}}\int _{B_{r}(x)}|f|.$

The Hardy-Littlewood maximal inequality states that there is some constant $C$ such that for all $f\in L^{1}(\mathbb {R} ^{n})$ and all $\alpha >0$,

$m\left(\{x:Hf(x)>\alpha \}\right)<{\frac {C}{\alpha }}\int _{\mathbb {R} ^{n}}|f|.$

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 ${\widehat {\cdot }}:L^{p}(\mathbb {R} ^{n})\to L^{p'}(\mathbb {R} ^{n})$ is a well-defined bounded operator when $1/p+1/p'=1$.

Heaviside

The Heaviside function is the function H on $\mathbb {R}$ such that $H(x)=1,\,x\geq 0$ and $H(x)=0,\,x<0$.

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 $\mathbb {C} ^{n}$ 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 $f$ is said to be integrable if $\int |f|\,d\mu <\infty$.

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 $(X,d_{X})$ and $(Y,d_{Y})$ is a bijection $f:X\to Y$ that preserves the metric: $d_{X}(x,x')=d_{Y}(f(x),f(x'))$ for all $x,x'\in X$.

L

Lebesgue differentiation theorem

The Lebesgue differentiation theorem states that for locally integrable $f\in L_{\text{loc}}^{1}(\mathbb {R} ^{n})$, the equalities

$\lim _{r\to 0}{\frac {1}{m(B_{r}(x))}}\int _{B_{r}(x)}|f(y)-f(x)|\,dy=0$

and

$\lim _{r\to 0}{\frac {1}{m(B_{r}(x))}}\int _{B_{r}(x)}f=f(x)$

hold for almost every $x$. The set where they hold is called the Lebesgue set of $f$, 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 $f$ between metric spaces is said to be Lipschitz continuous if $\sup _{x\neq y}{\frac {d(f(x),f(y))}{d(x,y)}}<\infty$.

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 $E\in \Sigma ,\ \ \mu (E)\geq 0.$
• $\mu (\varnothing )=0.$
• Countable additivity (or σ-additivity): For all countable collections $\{E_{k}\}_{k=1}^{\infty }$ of pairwise disjoint sets in Σ,

$$\mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k}).$$

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 $d:X\times X\to \mathbb {R} _{\geq 0}$, called a metric, such that (1) $d(x,y)=0$ iff $x=y$, (2) $d(x,y)\leq d(x,z)+d(z,y)$ for all $x,y,z\in X$, (3) $d(x,y)=d(y,x)$ for all $x,y\in X$.

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 $\lVert \cdot \rVert :V\to \mathbb {R}$ satisfying four axioms:

1. Non-negativity: for every $x\in V$,$\;\lVert x\rVert \geq 0$.
2. Positive definiteness: for every $x\in V$, $\;\lVert x\rVert =0$ if and only if $x$ is the zero vector.
3. Absolute homogeneity: for every scalar $\lambda$ and $x\in V$,$$\lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert$$
4. Triangle inequality: for every $x\in V$ and $y\in V$,$$\|x+y\|\leq \|x\|+\|y\|.$$

O

Oka

Oka's coherence theorem says the sheaf ${\mathcal {O}}_{\mathbb {C} ^{n}}$ 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 $\delta _{0}(x)=\int e^{2\pi ix\cdot \xi }\,d\xi .$

P

Paley

Paley–Wiener theorem

phase

The phase space to a configuration space $X$ (in classical mechanics) is the cotangent bundle $T^{*}X$ to $X$.

plurisubharmonic

A function $f$ on an open subset $U\subset \mathbb {C}$ is said to be plurisubharmonic if $t\mapsto f(z+tw)$ is subharmonic for $t$ in a neighborhood of zero in $\mathbb {C}$ and points $z,w$ in $U$.

Poisson

Poisson kernel

power series

A power series is informally a polynomial of infinite degree; i.e., $\sum _{n=1}^{\infty }a_{n}x^{n}$.

pseudoconex

A pseudoconvex set is a generalization of a convex set.

R

Radon measure

Let $X$ be a locally compact Hausdorff space and let $I$ be a positive linear functional on the space of continuous functions with compact support $C_{c}(X)$. Positivity means that $I(f)\geq 0$ if $f\geq 0$. There exist Borel measures $\mu$ on $X$ such that $I(f)=\int f\,d\mu$ for all $f\in C_{c}(X)$. A Radon measure on $X$ 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 $\mu$ on $X$ such that $I(f)=\int f\,d\mu$ for all $f\in C_{c}(X)$.

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 $z\mapsto \sum _{1}^{\infty }{\frac {1}{n^{z}}},\,\operatorname {Re} (z)>1$ (it's more traditional to write $s$ for $z$).

3.  The Riemann hypothesis, still a conjecture, says each nontrivial zero of the Riemann zeta function has real part equal to ${\frac {1}{2}}$.

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 $X$ is a map $\mathbb {N} \to X$.

series

A series is informally an infinite summation process $x_{1}+x_{2}+\cdots$. 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 $s_{n}:=x_{1}+\cdots +x_{n}$ 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 $X$ be a compact Hausdorff space and let $C(X,\mathbb {R} )$ have the uniform metric. One version of the Stone–Weierstrass theorem states that if ${\mathcal {A}}$ is a closed subalgebra of $C(X,\mathbb {R} )$ that separates points and contains a nonzero constant function, then in fact ${\mathcal {A}}=C(X,\mathbb {R} )$. 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 ${\mathcal {A}}$ is a subalgebra of $C(X,\mathbb {R} )$ that separates points and contains a nonzero constant function, then ${\mathcal {A}}$ is dense in $C(X,\mathbb {R} )$.

subanalytic

subanalytic.

subharmonic

A twice continuously differentiable function $f$ is said to be subharmonic if $\Delta f\geq 0$ where $\Delta$ 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 $\mathbb {N} {\overset {j}{\to }}\mathbb {N} {\overset {x}{\to }}X$ where $j$ is a strictly increasing injection and $x$ 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 $f_{n}:X\to E$ from a topological space to a normed space is said to converge uniformly to $f:X\to E$ if $\operatorname {sup} \|f_{n}-f\|\to 0$.

2.  A map between metric spaces is said to be uniformly continuous if for each $\epsilon >0$, there exist a $\delta >0$ such that $d(f(x),f(y))<\epsilon$ for all $x,y$ with $d(x,y)<\delta$.

V

Vitali covering lemma

The Vitali covering lemma states that if ${\mathcal {C}}$ is a collection of open balls in $\mathbb {R} ^{n}$ and

$c<m\left(\bigcup _{B\in {\mathcal {C}}}B\right),$

then there exists a finite number of balls $B_{1},\ldots ,B_{n}\in {\mathcal {C}}$ such that

$3^{n}\sum _{j=1}^{n}m(B_{j})>c.$

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

References

• Grauert, Hans; Remmert, Reinhold (1984). Coherent Analytic Sheaves. Springer.
• Halmos, Paul R. (1974) [1950], Measure Theory, Graduate Texts in Mathematics, vol. 18, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 978-0-387-90088-9, MR 0033869, Zbl 0283.28001
• Hörmander, Lars (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035.
• Hörmander, Lars (1966). An Introduction to Complex Analysis in Several Variables. Van Nostrand.
• Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. ISBN 9780070542358.
• Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill. ISBN 978-0-07-054234-1.
• Folland, Gerald B. (2007). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley.
• Jost, Jürgen (1998). Postmodern Analysis. Springer.
• Ahlfors, Lars V. (1978), Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (3rd ed.), McGraw-Hill

Further reading

• Semiclassical Microlocal Analysis（2020 Fall) by 王作勤 (wangzuoq)