Glossary of real and complex analysis

This is a glossary of concepts and results in real analysis and complex analysis in mathematics. In particular, it includes those in measure theory (as there is no glossary for measure theory in Wikipedia right now).

A

{{defn|1=An analytic continuation of a holomorphic function is a unique holomorphic extension of the function (on a connected open subset of $\mathbb{C}$ ).}}

{{defn|1=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

{{defn|no=1|1=A Borel measure is a measure whose domain is the Borel σ-algebra.}}

{{defn|no=2|1=The Borel σ-algebra on a topological space is the smallest σ-algebra containing all open sets.}}

{{defn|no=3|1=Borel's lemma says that a given formal power series, there is a smooth function whose Taylor series coincides with the given series.}}

{{defn|1=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 $a,b \in A$ .}}

{{defn|1=A bump function is a nonzero compactly-supported smooth function, usually constructed using the exponential function.}}

{{defn|1=A BV-function or a bounded variation is a function with bounded total variation.}}

C

{{defn|no=2|1=Caratheodory's criterion states a sufficient condition for Borel sets to be measurable.}}

{{defn|1=Cartwright's theorem gives a bounded for a p-valent entire function.}}

{{defn|no=1|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.}}

{{defn|no=5|1=The Cauchy principal value is, when possible, a number assigned to a function when the function is not integrable.}}

{{defn|no=6|1=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 $n, m \ge N$ .}}

{{defn|1=Cesàro summation is one way to compute a divergent series.}}

{{defn|1=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 $Y$ .}}

{{defn|The contour integral of a measurable function $f$ over a piece-wise smooth curve $\gamma : [0, 1] \to \mathbb{C}$ is $\int_{\gamma} f \, dz := \int_0^1 \gamma^*(f \, dz)$ .}}

{{defn|no=1|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.}}

{{defn|no=2|1=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 $d(x_n,x) < \epsilon$ .}}

{{defn|no=3|1=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.}}

{{defn|1=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.}}

{{defn|1=For sets $F \subset U$ , $F$ closed, $U$ open, a cutoff function is a function that is $1$ on $F$ and has support contained in $U$ . It’s usually required to be continuous or smooth.}}

D

{{defn|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 $\lim_{h \to 0} \| f(x + h) - f(x) - Th \|/\|h\| = 0$ .}}

{{defn|1=A map between normed space is differentiable at a point x if the derivative at x exists.}}

{{defn|1=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.}}

{{defn|1=The Dirac delta function $\delta_0$ on $\mathbb{R}^n$ is a distribution (so not exactly a function) given as $\langle \delta_0, \varphi \rangle = \varphi(0).$ }}

{{defn|1=A distribution is a type of a generalized function; precisely, it is a continuous linear functional on the space of test functions.}}

{{defn|1=A divergent series is a series whose partial sum does not converge. For example, $\sum_1^{\infty} \frac{1}{n}$ is divergent.}}

{{defn|1=The division conjecture of L. Schwartz (now a theorem) says a distribution divied by a real analyic function is again a distribution.}}

{{defn|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 $|f_n| \le g$ for some integrable function $g$ .}}

E

{{defn|An entire function is a holomorphic function whose domain is the entire complex plane.}}

{{defn|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

{{defn|no=1|1=The Fourier transform of a function $f$ on $\mathbb{R}^n$ is: (provided it makes sense)

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

{{defn|no=2|1=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

{{defn|1=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

{{defn|1=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 $\alpha > 0$ ,

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

{{defn|1=A function is harmonic if it satisfies the Laplace equation (in the distribution sense if the function is not twice differentiable).}}

{{defn|1=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 $1/p + 1/p' = 1$ .}}

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

{{defn|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.}}

{{defn|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

{{defn|1=A measurable function $f$ is said to be integrable if $\int |f| \, d\mu < \infty$ .}}

{{defn|no=1|1=The integral of the indicator function on a measurable set is the measure (volume) of the set.}}

{{defn|no=2|1=The integral of a measurable function is then defined by approximating the function by linear combinations of indicator functions.}}

{{defn|1=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

{{defn|The Lebesgue differentiation theorem states that for locally integrable $f \in L^1_{\text{loc}}(\mathbb R^n)$ , 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.

}}

{{defn|1=Levi's problem asks to show a pseudoconvex set is a domain of holomorphy.}}

{{defn|Liouville's theorem says a bounded entire function is a constant function.}}

{{defn|no=1|1=A map $f$ between metric spaces is said to be Lipschitz continuous if $\sup_{x \ne y} \frac{d(f(x), f(y))}{d(x, y)} < \infty$ .}}

{{defn|no=2|A map is locally Lipschitz continuous if it is Lipschitz continuous on each compact subset.}}

M

{{defn|The maximum principle says that a maximum value of a harmonic function in a connected open set is attained on the boundary.}}

{{defn|A measurable function is a structure-preserving function between measurable spaces in the sense that the preimage of any measurable set is measurable.}}

{{defn|A measurable set is an element of a {{math|σ}}-algebra.}}

{{defn|A measurable space consists of a set and a {{math|σ}}-algebra on that set which specifies what sets are measurable.}}

{{defn|A measure is a function on a measurable space that assigns to each measurable set a number representing its measure or size. Specifically, if {{math|X}} is a set and {{math|Σ}} is a {{math|σ}}-algebra on {{math|X}}, then a set-function {{math|μ}} from {{math|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 {{math|σ}}-additivity): For all countable collections $\{ E_k \}_{k=1}^\infty$ of pairwise disjoint sets in {{math|Σ}},

$\mu\left(\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty \mu(E_k).$ }}

{{defn|A measure space consists of a measurable space and a measure on that measurable space.}}

{{defn|1=A meromorphic function is an equivalence class of functions that are locally fractions of holomorphic functions.}}

{{defn|1=A metric space is a set {{math|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 $x,y \in X$ .}}

{{defn|1=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.}}

{{defn|Morera's theorem says a function is holomorphic if the integrations of it over arbitrary closed loops are zero.}}

N

{{defn|1=Nonsmooth analysis is a brach of mathematical analysis that concerns non-smooth functions like Lipschitz functions and has applications to optimization theory or control theory. Note this theory is generally different from distributional calculus, a calculus based on distributions.}}

{{defn|A normed vector space, also called a normed space, is a real or complex vector space {{math|V}} on which a norm is defined. A norm is a map $\lVert\cdot \rVert : V \to \mathbb R$ satisfying four axioms:

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

}}

O

{{defn|Oka's coherence theorem says the sheaf $\mathcal{O}_{\mathbb{C}^n}$ of holomorphic functions is coherent.}}

{{defn|1=An oscillatory integral can give a sense to a formal integral expression like $\delta_0(x) = \int e^{2 \pi i x \cdot \xi} \, d\xi.$ }}

P

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

{{defn|1=Plancherel's theorem says the Fourier transformation is a unitary operator.}}

{{defn|Plateau problem concerns the existence of a minimal surface.}}

{{defn|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 $U$ .}}

{{defn|A power series is informally a polynomial of infinite degree; i.e., $\sum_{n=0}^{\infty} a_n x^n$ .}}

{{defn|1=A pseudoconvex set is a generalization of a convex set.}}

R

{{defn|1=Rademacher's theorem says a locally Lipschitz function is differentiable almost everywhere.}}

{{defn|1=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 $f \in C_c(X)$ .}}

{{defn|1=A real-analytic function is a function given by a convergent power series.}}

{{defn|Rellich's lemma tells when an inclusion of a Sobolev space to another Sobolev space is a compact operator.}}

{{defn|no=1|The Riemann integral of a function is either the upper Riemann sum or the lower Riemann sum when the two sums agree.}}

{{defn|no=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 $z$ ).}}

{{defn|no=3|The Riemann hypothesis, still a conjecture, says each nontrivial zero of the Riemann zeta function has real part equal to $\frac{1}{2}$ .}}

S

{{defn|1=Sato's hyperfunction, a type of a generalized function.}}

{{defn|1=A Schwarz function is a function that is both smooth and rapid-decay.}}

{{defn|1=The notion of semianalytic is an analog of semialgebraic.}}

{{defn|A sequence on a set $X$ is a map $\mathbb{N} \to X$ .}}

{{defn|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.}}

{{defn|A σ-algebra on a set is a nonempty collection of subsets closed under complements, countable unions, and countable intersections.}}

{{defn|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 $C(X,\mathbb R)$ .}}

{{defn|1=A twice continuously differentiable function $f$ is said to be subharmonic if $\Delta f \ge 0$ where $\Delta$ is the Laplacian. The subharmonicity for a more general function is defined by a limiting process.}}

{{defn|1=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.}}

{{defn|no=1|The support of a function is the closure of the set of points where the function does not vanish.}}

{{defn|no=2|The support of a distribution is the support of it in the sense in sheaf theory.}}

T

{{defn|1=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.}}

{{defn|A tempered distribution is a distribution that extends to a continuous linear functional on the space of Schwarz functions.}}

{{defn|A test function is a compactly-supported smooth function; see also spaces of test functions and distributions.}}

U

{{defn|no=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 $\operatorname{sup} \| f_n - f \| \to 0$ .}}

{{defn|no=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

{{defn|1=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

{{defn|no=1|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.}}

References

{{cite book |last1=Grauert |first1=Hans | authorlink = Hans Grauert |last2=Remmert |first2=Reinhold |authorlink2= Reinhold Remmert |title=Coherent Analytic Sheaves |series=Grundlehren der mathematischen Wissenschaften |date=1984 |volume=265 |publisher=Springer |doi=10.1007/978-3-642-69582-7 |isbn=978-3-642-69584-1 |url=https://link.springer.com/book/10.1007/978-3-642-69582-7}}
{{Citation

| last = Halmos

| first = Paul R.

| author-link = Paul Halmos

| title = Measure Theory

| place = New York, Heidelberg, Berlin

| publisher = Springer-Verlag

| series = Graduate Texts in Mathematics

| volume = 18

| orig-date = 1950

| year = 1974

| isbn = 978-0-387-90088-9

| mr = 0033869

| zbl = 0283.28001

| url = https://archive.org/details/measuretheory00halm

}}

{{citation|mr=0717035|first=Lars |last= Hörmander|authorlink=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher=Springer |year=1983|isbn=3-540-12104-8 |doi=10.1007/978-3-642-96750-4}}.
{{cite book|first=Lars|last=Hörmander|author-link=Lars Hörmander|title=An Introduction to Complex Analysis in Several Variables|publisher=Van Nostrand | year=1966}}
{{cite book |author=Rudin, Walter | authorlink = Walter Rudin |date=1976 |title=Principles of Mathematical Analysis |title-link= Principles of Mathematical Analysis |edition=3rd |series=Walter Rudin Student Series in Advanced Mathematics |publisher=McGraw-Hill |isbn=9780070542358}}
{{cite book | last=Rudin | first=Walter | authorlink = Walter Rudin | title = Real and Complex Analysis (International Series in Pure and Applied Mathematics) | publisher=McGraw-Hill | year=1986 |isbn=978-0-07-054234-1}}
{{cite book |last1=Folland |first1=Gerald B. |authorlink=Gerald Folland |title=Real Analysis: Modern Techniques and Their Applications |date=2007 |publisher=Wiley |edition=2nd}}
{{cite book |last1=Jost |first1=Jürgen |authorlink=Jürgen Jost |title=Postmodern Analysis |date=1998 |publisher=Springer}}
{{citation|last=Ahlfors|first= Lars V.|authorlink=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|edition=3rd|series= International Series in Pure and Applied Mathematics|publisher= McGraw-Hill|year= 1978}}
{{cite book| last = Federer| first = Herbert|author-link1=Herbert Federer| title = Geometric measure theory| place= Berlin–Heidelberg–New York| publisher = Springer-Verlag| series = Die Grundlehren der mathematischen Wissenschaften| volume = 153| year = 1969| isbn = 978-3-540-60656-7| mr=0257325| zbl= 0176.00801 | doi=10.1007/978-3-642-62010-2}}

Glossary of real and complex analysis

A

B

C

D

E

F

G

H

I

L

M

N

O

P

R

S

T

U

V

W

References

Further reading