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). Also, the topics in algebraic analysis are included. However, the items in the theory of differential equations are not included. See also: list of real analysis topics, list of complex analysis topics and glossary of functional analysis.
0–9
1 + 2 + 3 + ⋯ = − 1 12 {\displaystyle 1+2+3+\cdots =-{\frac {1}{12}}}
It means the value of the zeta function at − 1 {\displaystyle -1} is − 1 / 12 {\displaystyle -1/12} . See 1 + 2 + 3 + 4 + ⋯.
A
Abel 1. Abel sum 2. Abel integral
absolute A series ∑ 0 ∞ a n {\displaystyle \sum _{0}^{\infty }a_{n}} is said to converge absolutely if ∑ 0 ∞ | a n | < ∞ . {\displaystyle \sum _{0}^{\infty }|a_{n}|<\infty .}
accumulation An accumulation point can mean either a limit point or a cluster point.
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 C {\displaystyle \mathbb {C} } ).
analytic sheaf analytic sheaf
archimedean The archimedean property of real numbers says: given two real numbers x , y {\displaystyle x,y} , if x > 0 {\displaystyle x>0} , then there exists an integer n > 0 {\displaystyle n>0} such that n x > y {\displaystyle nx>y} . argument principle argument principle
Ascoli Ascoli's theorem says that an equicontinous bounded sequence of functions on a compact subset of R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence with respect to the sup norm.
B
Bargmann Bargmann transform
Basel The Basel problem says ∑ 1 ∞ 1 n 2 = π 2 6 . {\displaystyle \sum _{1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}
Berezin Berezin integral
Bolzano The Bolzano-Weierstrass theorem says a bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. Today it is subsumed in the statement that a subset of a metric space is compact if and only if it is sequentially compact (i.e., every sequence has a convergent subsequence) and the Heine–Borel theorem.
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. 4. The Borel–Lebesgue lemma is another name for the Heine-Borel theorem.
bounded A subset A {\displaystyle A} of a metric space ( X , d ) {\displaystyle (X,d)} is bounded if there is some C > 0 {\displaystyle C>0} such that d ( a , b ) < C {\displaystyle d(a,b)<C} for all a , b ∈ A {\displaystyle a,b\in A} .
bump A bump function is a nonzero compactly-supported smooth function, usually constructed using the exponential function.
BV A BV-function or a bounded variation is a function with bounded total variation.
C
Calderón Calderón–Zygmund lemma
Cantor Cantor set.
capacity Capacity of a set is a notion in potential theory.
Carathéodory 1. Carathéodory's extension theorem 2. Carathéodory's criterion states a sufficient condition for Borel sets to be measurable.
Cartan Cartan's theorems A and B.
Cartwright Cartwright's theorem gives a bounded for a p-valent entire function.
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 {\displaystyle x_{n}} is called a Cauchy sequence if d ( x n , x m ) → 0 {\displaystyle d(x_{n},x_{m})\to 0} ; i.e., for each ϵ > 0 {\displaystyle \epsilon >0} , there is an N > 0 {\displaystyle N>0} such that d ( x n , x m ) < ϵ {\displaystyle d(x_{n},x_{m})<\epsilon } for all n , m ≥ N {\displaystyle n,m\geq N} . 7. Cauchy condensation test.
Cavalieri Cavalieri's principle.
Cesàro Cesàro summation is one way to compute a divergent series.
Clarke generalized derivative Clarke generalized derivative.
cluster 1. A cluster point of a net x α {\displaystyle x_{\alpha }} (or a sequence) is a point in ⋂ α { x β ∣ β ≳ α } ¯ {\displaystyle \bigcap _{\alpha }{\overline {\{x_{\beta }\mid \beta \gtrsim \alpha \}}}} . The notion gives a convenient criterion for compactness: a space (resp. a metric space) is compact if and only if each net (resp. sequence) has a cluster point. 2. For a first countable space, a point is a cluster point of a sequence x i {\displaystyle x_{i}} if and only if there is a subsequence converging to that point.
complex 1. A complex number is an element in C = R [ x ] / ( x 2 + 1 ) {\displaystyle \mathbb {C} =\mathbb {R} [x]/(x^{2}+1)} , the quotient ring of a polynomial ring, where the image of the indeterminate x {\displaystyle x} is denoted by i {\displaystyle i} . As a set, C {\displaystyle \mathbb {C} } can be identified with R 2 {\displaystyle \mathbb {R} ^{2}} and that gives a topology on it. 2. complex logarithm. 3. Given a function f {\displaystyle f} on an open set U ⊂ C {\displaystyle U\subset \mathbb {C} } , the complex derivative of it is, if any, the limit lim C ∋ h → 0 f ( z + h ) − f ( z ) h {\displaystyle \lim _{\mathbb {C} \ni h\to 0}{\frac {f(z+h)-f(z)}{h}}} .
continuous A function f : X → Y {\displaystyle f:X\to Y} between metric spaces ( X , d X ) {\displaystyle (X,d_{X})} and ( Y , d Y ) {\displaystyle (Y,d_{Y})} is continuous if for any convergent sequence x n → x {\displaystyle x_{n}\to x} in X {\displaystyle X} , we have f ( x n ) → f ( x ) {\displaystyle f(x_{n})\to f(x)} in Y {\displaystyle Y} .
contour The contour integral of a measurable function f {\displaystyle f} over a piece-wise smooth curve γ : [ 0 , 1 ] → C {\displaystyle \gamma :[0,1]\to \mathbb {C} } is ∫ γ f d z := ∫ 0 1 γ ∗ ( f d z ) {\displaystyle \int _{\gamma }f\,dz:=\int _{0}^{1}\gamma ^{*}(f\,dz)} .
converge 1. A sequence x n {\displaystyle x_{n}} in a topological space is said to converge to a point x {\displaystyle x} if for each open neighborhood U {\displaystyle U} of x {\displaystyle x} , the set { n ∣ x n ∉ U } {\displaystyle \{n\mid x_{n}\not \in U\}} is finite. 2. A sequence x n {\displaystyle x_{n}} in a metric space is said to converge to a point x {\displaystyle x} if for all ϵ > 0 {\displaystyle \epsilon >0} , there exists an N > 0 {\displaystyle N>0} such that for all n > N {\displaystyle n>N} , we have d ( x n , x ) < ϵ {\displaystyle d(x_{n},x)<\epsilon } . 3. A series x 1 + x 2 + ⋯ {\displaystyle x_{1}+x_{2}+\cdots } on a normed space (e.g., R n {\displaystyle \mathbb {R} ^{n}} ) is said to converge if the sequence of the partial sums s n := ∑ 1 n x j {\displaystyle s_{n}:=\sum _{1}^{n}x_{j}} converges.
convolution The convolution f ∗ g {\displaystyle f*g} of two functions on a convex set is given by
( f ∗ g ) ( x ) = ∫ f ( y − x ) g ( y ) d y , {\displaystyle (f*g)(x)=\int f(y-x)g(y)\,dy,}
provided the integration converges.
Cousin Cousin problems.
critical critical point.
cutoff For sets F ⊂ U {\displaystyle F\subset U} , F {\displaystyle F} closed, U {\displaystyle U} open, a cutoff function is a function that is 1 {\displaystyle 1} on F {\displaystyle F} and has support contained in U {\displaystyle U} . It’s usually required to be continuous or smooth.
D
de Branges de Branges's theorem.
Dedekind A Dedekind cut is one definition of a real number. By definition, it is a nonempty proper lower subset α {\displaystyle \alpha } of Q {\displaystyle \mathbb {Q} } that has no maximal element, where lower means it contains { q ∈ Q ∣ q < p } {\displaystyle \{q\in \mathbb {Q} \mid q<p\}} for each p {\displaystyle p} in α {\displaystyle \alpha } . For example, 2 = { p ∈ Q ∣ p < 0 or p 2 < 2 } {\displaystyle {\sqrt {2}}=\{p\in \mathbb {Q} \mid p<0{\text{ or }}p^{2}<2\}} .
derivative Given a map f : E → F {\displaystyle f:E\to F} between normed spaces, the derivative of f {\displaystyle f} at a point x is a (unique) linear map T : E → F {\displaystyle T:E\to F} such that lim h → 0 ‖ f ( x + h ) − f ( x ) − T h ‖ / ‖ h ‖ = 0 {\displaystyle \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 Differentiation under the integral sign
Dini Dini's theorem.
Dirac 1. The Dirac delta function δ 0 {\displaystyle \delta _{0}} on R n {\displaystyle \mathbb {R} ^{n}} is a distribution (so not exactly a function) given as ⟨ δ 0 , φ ⟩ = φ ( 0 ) . {\displaystyle \langle \delta _{0},\varphi \rangle =\varphi (0).}
2. A Dirac sequence.[1]
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, ∑ 1 ∞ 1 n {\displaystyle \sum _{1}^{\infty }{\frac {1}{n}}} is divergent.
division conjecture The division conjecture of L. Schwartz (now a theorem) says a distribution divided by a real analytic function is again a distribution.
dominated Lebesgue's dominated convergence theorem says ∫ f n d μ {\displaystyle \int f_{n}\,d\mu } converges to ∫ f d μ {\displaystyle \int f\,d\mu } if f n {\displaystyle f_{n}} is a sequence of measurable functions such that f n {\displaystyle f_{n}} converges to f {\displaystyle f} pointwise and | f n | ≤ g {\displaystyle |f_{n}|\leq g} for some integrable function g {\displaystyle g} .
E
e Euler's number. One definition is through the series representation of the exponential function; namely, e = ∑ 0 ∞ 1 n ! {\displaystyle e=\sum _{0}^{\infty }{\frac {1}{n!}}} . Another is through natural logarithm; namely, log ( e ) = 1 {\displaystyle \log(e)=1} .
edge Edge-of-the-wedge theorem.
Egoroff Egoroff's theorem.
entire An entire function is a holomorphic function whose domain is the entire complex plane.
equicontinuous A set S {\displaystyle S} of maps between fixed metric spaces is said to be equicontinuous if for each ϵ > 0 {\displaystyle \epsilon >0} , there exists a δ > 0 {\displaystyle \delta >0} such that sup f ∈ S d ( f ( x ) , f ( y ) ) < ϵ {\displaystyle \sup _{f\in S}d(f(x),f(y))<\epsilon } for all x , y {\displaystyle x,y} with d ( x , y ) < δ {\displaystyle d(x,y)<\delta } . A map f {\displaystyle f} is uniformly continuous if and only if { f } {\displaystyle \{f\}} is equicontinuous.
Euler The Euler–Maclaurin formula.
exponential The exponential function is the function z ↦ e z {\displaystyle z\mapsto e^{z}} on the complex plane, where e {\displaystyle e} is Euler's number. If the number e {\displaystyle e} is defined through the exponential function, then the exponential function is defined more directly as: z ↦ ∑ 1 ∞ z n n ! {\displaystyle z\mapsto \sum _{1}^{\infty }{\frac {z^{n}}{n!}}} .
F
Fatou Fatou's lemma
finite interesection property Given a topological space X {\displaystyle X} , a family F {\displaystyle F} of closed subsets of X {\displaystyle X} is said to have the finite intersection property if each finite subset of F {\displaystyle F} has nonempty intersection. Then saying X {\displaystyle X} is compact can be restated as: each family of closed subsets of X {\displaystyle X} with the finite intersection property has nonempty intersection.
filter 1. A filter F {\displaystyle F} on a set X {\displaystyle X} is a proper subset of the power set of X {\displaystyle X} such that (upper) if S {\displaystyle S} is in F {\displaystyle F} , each subset of X {\displaystyle X} containing S {\displaystyle S} is also in F {\displaystyle F} and (downward directed) the intersection of each finite subset of F {\displaystyle F} is in F {\displaystyle F} . Its role is similar to that of nets but in analysis, nets are more commonly used. 2. Given a net x α {\displaystyle x_{\alpha }} , there is the filter F {\displaystyle F} determined by it; namely, the filter generated by the tails { x β ∣ β ≳ α } {\displaystyle \{x_{\beta }\mid \beta \gtrsim \alpha \}} . Then for example, x α {\displaystyle x_{\alpha }} converges to a point x {\displaystyle x} if and only if F {\displaystyle F} converges to x {\displaystyle x} (meaning F {\displaystyle F} contains every neighborhood of x {\displaystyle x} ). Conversely, given a filter, we can choose a net associated to it so that all the associated nets determine the original filter.
first A first countable space is a topological space in which each point x {\displaystyle x} has a decreasing sequence of neighborhoods x ∈ ⋯ ⊂ U 2 ⊂ U 1 ⊂ U 0 {\displaystyle x\in \cdots \subset U_{2}\subset U_{1}\subset U_{0}} such that each neighborhood of x {\displaystyle x} contains some U n {\displaystyle U_{n}} . An important property of such a space is that a point is in the closure of a set E {\displaystyle E} if and only if there is a sequence in E {\displaystyle E} that converges to that point.
Fock Fock space
Fourier 1. The Fourier transform of a function f {\displaystyle f} on R n {\displaystyle \mathbb {R} ^{n}} is: (provided it makes sense)
f ^ ( ξ ) = ∫ f ( x ) e − 2 π i x ⋅ ξ d x . {\displaystyle {\widehat {f}}(\xi )=\int f(x)e^{-2\pi ix\cdot \xi }\,dx.}
2. The Fourier transform f ^ {\displaystyle {\widehat {f}}} of a distribution f {\displaystyle f} is ⟨ f ^ , φ ⟩ = ⟨ f , φ ^ ⟩ {\displaystyle \langle {\widehat {f}},\varphi \rangle =\langle f,{\widehat {\varphi }}\rangle } . For example, δ 0 ^ = 1 {\displaystyle {\widehat {\delta _{0}}}=1} (Fourier's inversion formula).
Fubini Fubini's theorem computes an integral as iterated integrals.
G
Gamma Gamma function.
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.
germ The germ of a function at a point p {\displaystyle p} is the equivalence class of functions (of some class) on neighborhoods of the point, where f ∼ g {\displaystyle f\sim g} if the restrictions of f , g {\displaystyle f,g} are the same on some neighborhood of the point.
Grandi Grandi's series is the series ∑ 0 ∞ ( − 1 ) n {\displaystyle \sum _{0}^{\infty }(-1)^{n}} . It is divergent but some summation methods can be used to show its value is 1 / 2 {\displaystyle 1/2} .
Grauert 1. Hans Grauert. 2. Grauert's approximation theorem.
H
Hardy-Littlewood maximal inequality The Hardy-Littlewood maximal function of f ∈ L 1 ( R n ) {\displaystyle f\in L^{1}(\mathbb {R} ^{n})} is
H f ( x ) := sup r > 0 1 m ( B r ( x ) ) ∫ B r ( x ) | f | . {\displaystyle 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 {\displaystyle C} such that for all f ∈ L 1 ( R n ) {\displaystyle f\in L^{1}(\mathbb {R} ^{n})} and all α > 0 {\displaystyle \alpha >0} ,
m ( { x : H f ( x ) > α } ) < C α ∫ R n | f | . {\displaystyle 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 ⋅ ^ : L p ( R n ) → L p ′ ( R n ) {\displaystyle {\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 {\displaystyle 1/p+1/p'=1} .
Heaviside The Heaviside function is the function H on R {\displaystyle \mathbb {R} } such that H ( x ) = 1 , x ≥ 0 {\displaystyle H(x)=1,\,x\geq 0} and H ( x ) = 0 , x < 0 {\displaystyle H(x)=0,\,x<0} .
Heine 1. The Heine–Borel theorem says a subset of R n {\displaystyle \mathbb {R} ^{n}} is compact if and only if it is closed and bounded. 2. The above theorem follows from a more general result: a metric space is compact if and only if it is complete and totally ordered, since a bounded set in a Euclidean space is totally bounded.
Hermite Hermite polynomial
Hilbert space 1. 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. 2. Hilbert transform.
holomorphic function A function defined on an open subset of C n {\displaystyle \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).
hypoelliptic A hypoelliptic operator is an operator for which the elliptic regularity holds.
I
infinitesimal An infinitesimal is a "number" that is greater then zero but is smaller than any positive real number; in particular, it is not a real number.
integrable A measurable function f {\displaystyle f} is said to be integrable if ∫ | f | d μ < ∞ {\displaystyle \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.
inverse The inverse function theorem gives a necessary and sufficient condition for a function to be injective. Note it only gives an "inverse function" on the image of the function.
isolated An isolated point of a set is a point that is not a limit point of the set.
isometry An isometry between metric spaces ( X , d X ) {\displaystyle (X,d_{X})} and ( Y , d Y ) {\displaystyle (Y,d_{Y})} is a bijection f : X → Y {\displaystyle f:X\to Y} that preserves the metric: d X ( x , x ′ ) = d Y ( f ( x ) , f ( x ′ ) ) {\displaystyle d_{X}(x,x')=d_{Y}(f(x),f(x'))} for all x , x ′ ∈ X {\displaystyle x,x'\in X} .
J
jet jet space.
L
Laplace Laplace transform.
Lebesgue differentiation theorem The Lebesgue differentiation theorem states that for locally integrable f ∈ L loc 1 ( R n ) {\displaystyle f\in L_{\text{loc}}^{1}(\mathbb {R} ^{n})} , the equalities
lim r → 0 1 m ( B r ( x ) ) ∫ B r ( x ) | f ( y ) − f ( x ) | d y = 0 {\displaystyle \lim _{r\to 0}{\frac {1}{m(B_{r}(x))}}\int _{B_{r}(x)}|f(y)-f(x)|\,dy=0}
and
lim r → 0 1 m ( B r ( x ) ) ∫ B r ( x ) f = f ( x ) {\displaystyle \lim _{r\to 0}{\frac {1}{m(B_{r}(x))}}\int _{B_{r}(x)}f=f(x)}
hold for almost every x {\displaystyle x} . The set where they hold is called the Lebesgue set of f {\displaystyle f} , and points in the Lebesgue set are called Lebesgue points.
Lebesgue dominated convergence theorem Lebesgue dominated convergence theorem
Lebesgue 1. Lebesgue integral. 2. Lebesgue measure. 3. Given an open cover U {\displaystyle {\mathcal {U}}} of a metric space X {\displaystyle X