complete

en-adj

complete (comparative more complete or completer, superlative most complete or completest)

more complete (comparative)
completer (comparative)
most complete (superlative)
completest (superlative)
compleat (alternative archaic)

From Middle English compleet (“full, complete”), borrowed from Old French complet or Latin completus, past participle of compleō (“I fill up, I complete”) (whence also complement, compliment), from com- + pleō (“I fill, I fulfill”) (whence also deplete, replete, plenty), ultimately from Proto-Indo-European *pleh₁- (“to fill”) (English full).

пълен

of a proof system: such that any semantically valid formula is also provable

complete androgen insensitivity syndrome

QMA arises naturally in the study of quantum computation, and it also has a complete problem, Local Hamiltonian, which is a generalization of k-SAT.

2007, Yi-Kai Liu, The Complexity of the Consistency and N-representability Problems for Quantum States, page 17:

BPP behaves differently in some ways from other classes we have seen. For example, we know of no complete languages for BPP.

2009, Sanjeev Arora, Boaz Barak, Computational Complexity: A Modern Approach, page 137:

(computing theory, of a problem) That is in a given complexity class and is such that every other problem in the class can be reduced to it (usually in polynomial time or logarithmic space).


spectrum

From Latin spectrum (“appearance, image, apparition”), from speciō (“look at, view”). Doublet of specter. See also scope.

bor

(mathematics, functional analysis) Of a bounded linear operator A, the set of scalar values λ such that the operator A—λI, where I denotes the identity operator, does not have a bounded inverse; intended as a generalisation of the linear algebra sense.


functional

/ˈfʌŋkt͡ʃənəl/

(mathematics, functional analysis) A scalar-valued linear function on a vector space.


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(functional analysis, of a Hilbert space operator) Which commutes with its adjoint.


(linear algebra, functional analysis) Dual space.


norm

From Latin norma (“a carpenter's square, a rule, a pattern, a precept”). Doublet of norma.

норма

math: function that maps vectors to non-negative reals

(philosophy, computer science) A sentence with non-descriptive meaning, such as a command, permission, or prohibition.

(mathematics) A function which satisfies a particular set of formal conditions, created to generalize the notion of the length of a vector. Formally, a real-valued function on a vector space, generally denoted v↦|v| or v↦‖v‖, that satisfies the following properties:

given a scalar k, ‖kv‖=|k|·‖v‖, where |k| is the absolute value of k;

given two vectors v,w, ‖v+w‖<‖v‖+‖w‖ (the triangle inequality).

A quaternion algebra (a,b) over ksplits if and only if b is a norm from the field extension k(#92;sqrt#123;a#125;)#47;k, i.e. if and only if there is some x in k(#92;sqrt#123;a#125;) which has field norm exactly equal to b.

(mathematics) Any of several generalizations of the above: a field norm, ideal norm, etc.

(algebra) An element of the image of some (generalized) norm, the element then said to be from the norm in question, or from the structure which gave rise to the norm.


concave

en-adj

concave (comparative more concave, superlative most concave)

more concave (comparative)
most concave (superlative)

From Middle English concave, from Old French concave, from Latin concavus.

(functional analysis, not comparable, of a real-valued function on the reals) Satisfying the property that all segments connecting two points on the function's graph lie below the function.


convex

convex (comparative more convex, superlative most convex)

more convex (comparative)
most convex (superlative)

Borrowed from Middle French convexe, from Latin convexus (“arched”).

(functional analysis, not comparable, of a real-valued function on the reals) having an epigraph that is a convex set.


Hilbert space

en-noun

Hilbert space (countable and uncountable, plural Hilbert spaces)

Named after German mathematician David Hilbert (1862–1943).

named-after

(functional analysis) A generalized Euclidean space in which mathematical functions take the place of points; crucial to the understanding of quantum mechanics and other applications.


orthonormal

LL-Q1860 (eng)-Flame, not lame-orthonormal.wav

Blend of orthogonal + normal, equivalent to ortho- + normal.

Given a set of linearly independent vectors, it is often useful to convert them into an orthonormal set of vectors.

(Can we date this quote?), “Lesson 10”, in University College London, 6.4 Gram-Schmidt Process, page 2:

(functional analysis, geometry, linear algebra) Of a set of vectors, both orthogonal and normalized.

linear transformation
linear transformation

(functional analysis, geometry, linear algebra) Of a linear transformation: that preserves both angles and lengths.


semi-norm

semi-norms (plural)
seminorm (alternative)

mathematical analysis
mathematical analysis

(mathematical analysis) A function denoted ∥v∥ that maps a vector v to a non-negative value such that ∥cv∥ = |c|.∥v∥, where c is a scalar, and ∥v + w∥ ≤ ∥v∥ + ∥w∥ (the triangle inequality); the condition that ∥v∥ = 0 implies that v = 0 is not required, but when it holds, the semi-norm is a norm.

полунорма

a function that maps a vector to a non-negative value


convolution

en-noun

convolution (countable and uncountable, plural convolutions)

Borrowed from Latin convolutus (“to roll together”), past participle of convolvere, from con- + volvere (“to roll”), with the suffix -tion. Equivalent to convolute + -ion. The mathematical sense is a semantic calque from Faltung.

1934, Aurel Wintner,
The proper method in dealing with distribution functions and their convolutions (“Faltungen”) is the method of Fourier transforms, first applied systematically by Levy in his book on the calculus of probability.

Linear convolution is one of the most frequent computations carried out in digital signal processing (DSP).

1997, Richard Tolimieri, Myoung An, Chao Lu, Algorithms for Discrete Fourier Transform and Convolution, 2nd edition, Springer, page 101:

In Chapter 11 we considered H-convolutions of generalized H-transforms. These convolutions are bilinear, commutative and associative operations[…].

1994, Semen B. Yakubovich, Yurii F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Springer, page 183:

(mathematics, functional analysis) A mathematical operation on two functions that produces a third that expresses how the shape of one is modified by the other; the integral of the product of the two functions after one is reflected about the y-axis and shifted along the x-axis.

конволюция

mathematics: mathematical operation on two functions


Pythagorean theorem

the Pythagorean theorem (canonical)
Pythagoras' theorem (alternative)
theorem of Pythagoras (alternative)

Named after Pythagoras (from Ancient Greek Πυθαγόρας (Puthagóras)), the Greek mathematician and philosopher who by tradition is credited with the theorem’s discovery and proof.

(functional analysis) A generalization of the Pythagorean theorem for Euclidean triangles to Hilbert spaces.


orthogonal function

orthogonal function (plural orthogonal functions)

(functional analysis) Either of a pair of functions whose inner product is zero


orthonormal function

orthonormal function (plural orthonormal functions)

(functional analysis) An orthogonal function whose inner product with itself is unity


functional analysis

(functional analysis) The branch of mathematics dealing with infinite-dimensional vector spaces, whose elements are actually functions, as well as generalizations such as Banach spaces and Hilbert spaces.

functional analysis
simple:functional analysis


Banach space

Named after Polish mathematician Stefan Banach (1892–1945).

named-after

1962 [Prentice-Hall], Kenneth Hoffman, Banach Spaces of Analytic Functions, 2007, Dover, page 138,
Before taking up the extreme points for H¹ and H ᪲, let us make a few elementary observations about the unit ball Σ in the Banach space X.

Already in these cases there is convergence in Banach spaces that are not only infinite-dimensional but nonseparable.

1992, R. M. Dudley, M. G. Hahn, James Kuelbs, editors, Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference, Springer, page ix:

(functional analysis) A normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.


self-adjoint

(functional analysis) Being a Hilbert space operator which is Hermitian and also whose eigenvectors span the entire Hilbert space.


Cauchy sequence

en-noun

Cauchy sequence (plural Cauchy sequences)

Named after French mathematician Augustin-Louis Cauchy (1789–1857), who made pioneering contributions to analysis.

However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.

1955, [Van Nostrand], John L. Kelley, General Topology, Springer, published 1975, page 174:

In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. In the general case, however, this is not so. If a metric space does have the property that every Cauchy sequence converges, the space is called a complete metric space.

2000, George Bachman, Lawrence Narici, Functional Analysis, page 52:

Cantor first redefined Cauchy sequences using rational numbers only.[…]Cantor's idea was to define the real number line as the collection of all (rational) Cauchy sequences.

2012, David Applebaum, Limits, Limits Everywhere: The Tools of Mathematical Analysis, Oxford University Press, page 153:

(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every ϵ>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<ϵ.


バナッハ空間

バナッハ空間 (canonical)
Banahha kūkan (romanization)

(mathematical analysis) Banach space - a normed vector space which is complete, in the sense that Cauchy sequences converge


PVR

(functional analysis) Initialism of peak-to-valley ratio.


liniowy

pl-adj

liniowy (comparative bardziej liniowy, superlative najbardziej liniowy, derived adverb liniowo)

bardziej liniowy (comparative)
najbardziej liniowy (superlative)
liniowy (masculine nominative singular)
liniowa (feminine nominative singular)
liniowe (neuter nominative singular)
liniowi (nominative plural virile)
liniowe (nominative plural)
liniowego (genitive masculine singular)
liniowej (feminine genitive singular)
liniowego (genitive neuter singular)
liniowych (genitive plural)
liniowych (genitive plural virile)
liniowemu (dative masculine singular)
liniowej (dative feminine singular)
liniowemu (dative neuter singular)
liniowym (dative plural)
liniowym (dative plural virile)
liniowego (accusative animate masculine singular)
liniowy (accusative inanimate masculine singular)
liniową (accusative feminine singular)
liniowe (accusative neuter singular)
liniowych (accusative plural virile)
liniowe (accusative plural)
liniowym (instrumental masculine singular)
liniową (feminine instrumental singular)
liniowym (instrumental neuter singular)
liniowymi (instrumental plural)
liniowymi (instrumental plural virile)
liniowym (locative masculine singular)
liniowej (feminine locative singular)
liniowym (locative neuter singular)
liniowych (locative plural)
liniowych (locative plural virile)

(calculus, functional analysis, linear algebra, relational) linear (of first-degree polynomial)


linear operator

en-noun

linear operator (plural linear operators)

(mathematics, functional analysis) An operator L such that for functions f and g and scalar λ, L (f + g) = L f + L g and L λf = λ L f.


extensionality

extensionality (usually uncountable, plural extensionalities)

ekstensjonalność

the principle of distinguishing only by externally observable properties

(mathematics, functional analysis) The principle that functions are equal if and only if they operate on the same domain and for any given element of the domain the result of each of the functions is the same.


orthogonal complement

orthogonal complement (plural orthogonal complements)

(linear algebra, functional analysis) The set of all vectors which are orthogonal to a given set of vectors.


(functional analysis) Free additive convolution.


(functional analysis) Free multiplicative convolution


cotype

Rademacher distribution
Rademacher distribution

(functional analysis) A Banach space (X,‖·‖), such that, given a sequence of independent random variables from a Rademacher distribution, (εᵢ), and a rank q in the range from 2 to infinity, there exists a finite constant C≥1 such that for all finite sequences (x_i)∈X, 𝔼_ε[‖∑ limits ᵢ₌₁ⁿεᵢx_i‖^q]≥1/(C^q)(∑ limits ᵢ₌₁ⁿ‖x_i‖^q), if;2≤q<∞ or 𝔼_ε[‖∑ limits ᵢ₌₁ⁿεᵢx_i‖]≥1/C sup ‖x_i‖, if;q=∞ .


espaço de Hilbert

pt-noun

espaço de Hilbert m (plural espaços de Hilbert)

(functional analysis) Hilbert space (space in which functions take the place of points)


ピタゴラスの定理

ja-noun

^ピタゴラス の ていり

ピタゴラスの定理 (canonical)
Pitagorasu no teiri (romanization)

ピタゴラス (Pitagorasu, “Pythagoras”) + の (no, “'s”) + 定理 (teiri, “theorem”)

com

(geometry, functional analysis) the Pythagorean theorem


Krein-Milman theorem

A less general version was originally proved by Mark Krein and David Milman.

(mathematics, functional analysis) A proposition about convex sets in topological vector spaces.


théorème de Pythagore

LL-Q150 (fra)-WikiLucas00-théorème de Pythagore.wav


Triebel-Lizorkin space

en-noun

Triebel-Lizorkin space (plural Triebel-Lizorkin spaces)

Named after Hans Triebel and Petr Ivanovich Lizorkin.

(functional analysis) A generalization of many standard function spaces such as Lᵖ spaces and Sobolev spaces.


関数解析学

ja-noun

関(かん)数(すう)解(かい)析(せき)学(がく) • (kansū kaisekigaku)

関数解析学 (canonical)
kansū kaisekigaku (romanization)

[kã̠ɰ̃sɨː ka̠ise̞kʲiɡa̠kɯ̟]

関数 (kansū, “function”) + 解析学 (kaisekigaku, “analysis”)


topological vector space

en-noun

topological vector space (plural topological vector spaces)

(functional analysis) A vector space over a topological field endowed with a topology (often the real or complex numbers with standard topology) such that vector addition and scalar multiplication are continuous functions.

linear topological space

vector space with topology such that addition and scalar multiplication are continuous


konkava

eo-head

konkava (accusative singular konkavan, plural konkavaj, accusative plural konkavajn)

konkavan (accusative singular)
konkavaj (plural)
konkavajn (accusative plural)


Dirichlet energy

en-noun

Dirichlet energy (plural Dirichlet energies)

Dirichlet energies (plural)
Dirichlet's energy (alternative)

Named after German mathematician Peter Gustav Lejeune Dirichlet (1805–1859), who made significant contributions to the theory of Fourier series.

named-after

Variational principles play an important role in both geometry and physics, and one of the key problems with applications in both fields is the variational problem associated to the Dirichlet energy of maps between Riemannian manifolds.

2005, Roger Moser, Partial Regularity for Harmonic Maps and Related Problems, World Scientific, page 1:

The Dirichlet energy of a function f#92;inW#123;1,2#125; can be recovered, moreover, as the energy of the composition #92;xi#95;#123;BW#125;#92;circf, where #92;xi#95;#123;BW#125; is the biLipschitz embedding in Corollary 2.2 (compare with Theorem 2.4).

2011, Camillo De Lellis, Emanuele Nunzio Spadaro, Q-valued Functions Revisited, American Mathematical Society, page 28:

Since the Dirichlet energy is weakly lower semicontinuous and strongly continuous, the linear lower-order terms are weakly continuous on H#95;D¹(#92;Omega), and since the finite element spaces are dense in H#95;D¹(#92;Omega), we verify that I#95;h#92;to#92;GammaI as h#92;to 0.

2015, Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, page 89:

(mathematical analysis, functional analysis, Fourier analysis) A quadratic functional which, given a real function defined on an open subset of ℝⁿ, yields a real number that is a measure of how variable said function is.

functional analysis; Fourier analysis; functional analysis; Fourier analysis

енергия на Дирихле

functional that maps a function to a real number representing its variability


funkcja liniowa

funkcja liniowa (nominative singular)
funkcje liniowe (nominative plural)
funkcji liniowej (genitive singular)
funkcji (genitive plural)
funkcyj  liniowych (archaic genitive plural)
funkcji liniowej (dative singular)
funkcjom liniowym (dative plural)
funkcję liniową (accusative singular)
funkcje liniowe (accusative plural)
funkcją liniową (instrumental singular)
funkcjami liniowymi (instrumental plural)
funkcji liniowej (locative singular)
funkcjach liniowych (locative plural)
funkcjo liniowa (singular vocative)
funkcje liniowe (plural vocative)

(functional analysis, linear algebra) linear function (linear map)


superproperty

For our second principle we need the concept of superproperty. If (P) is a property of infinite dimensional Banach spaces, e.g. reflexivity, cotype, etc. we say that the infinite dimensional Banach space E has super (P) if any infinite dimensional Banach space F, F#92;hookrightarrowE, has property (P). If (P) = super (P), i.e. if a Banach space has super (P) if and only if it has (P) we call (P) a superproperty.

1999, Seán Dineen, “Polynomials”, in Complex Analysis on Infinite Dimensional Spaces (Springer Monographs in Mathematics), London: Springer, →ISBN, page 56:

(functional analysis) A property of an infinite dimensional Banach space E such that for any infinite dimensional Banach space F, F↪E, has that same property.


konvolusi

Borrowed from Dutch convolutie (“convolution”).

bor+

(mathematics, functional analysis) a mathematical operation on two functions that produces a third that expresses how the shape of one is modified by the other; the integral of the product of the two functions after one is reflected about the y-axis and shifted along the x-axis


infrabarrelled

topological vector space
topological vector space#English

(functional analysis) Describing a locally convex topological vector space for which every bounded barrel is a neighborhood of the origin.


bornivore

(functional analysis) A bornivorous set.


bornivorous

(functional analysis) Describing a subset of a real or complex vector space X, having an associated vector bornology B, that absorbs every element of B.


Tonelli's theorem

Named after the Italian mathematician Leonida Tonelli.

(functional analysis) A fundamental result on the weak lower semicontinuity of nonlinear functionals on Lᵖ spaces.