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114 relations: Absolute value, Alain Connes, Andrey Kolmogorov, Avner Friedman, Axiom of choice, École Centrale Paris, Baire category theorem, Banach space, Barry Simon, Basis (linear algebra), Boolean prime ideal theorem, Bounded operator, C*-algebra, Calculus of variations, Cardinal number, Combinatorics, Compact space, Complete metric space, Complex number, Continuous function, Continuous linear operator, Counting measure, Derivative, Differential equation, Dimension (vector space), Dual space, Eigenfunction, Ergodic theory, Erwin Kreyszig, Essential supremum and essential infimum, Fourier transform, Fréchet derivative, Fréchet space, Frigyes Riesz, Function space, Functional (mathematics), George Mackey, Gerald Teschl, Haïm Brezis, Hahn–Banach theorem, Hans Hahn (mathematician), Hausdorff space, Higher-order function, Hilbert space, Hugo Steinhaus, Integer, Integral, Integral equation, Invariant subspace, Invariant subspace problem, ..., Isometry, Isomorphism, Israel Gelfand, Jacob T. Schwartz, Jacques Hadamard, Jean Bourgain, John B. Conway, Juliusz Schauder, Kōsaku Yosida, Law of large numbers, Linear algebra, Linear form, Linear map, Linear subspace, List of functional analysis topics, Lp space, Lwów School of Mathematics, Mathematical analysis, Mathematical formulation of quantum mechanics, Mathematical physics, Maurice René Fréchet, Measurable function, Measure (mathematics), Measure space, Michael C. Reed, Multiplication operator, Nelson Dunford, Noncommutative geometry, Normal operator, Normed vector space, Open and closed maps, Open mapping theorem (functional analysis), Open set, Operator algebra, Operator theory, Orthonormal basis, Paul Lévy (mathematician), Peter Lax, Poland, Probability, Quantum mechanics, Real number, Representation theory, Schauder basis, Separable space, Sergei Fomin, Sobolev, Spectral theorem, Spectral theory, Stefan Banach, Sublinear function, Surjective function, Topological group, Topological ring, Topological space, Topological vector space, Uniform boundedness principle, Unitary operator, University of Colorado Colorado Springs, Up to, Vector space, Vito Volterra, Walter Rudin, Zorn's lemma. Expand index (64 more) »
Absolute value
In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.
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Alain Connes
Alain Connes (born 1 April 1947) is a French mathematician, currently Professor at the Collège de France, IHÉS, Ohio State University and Vanderbilt University.
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Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov (a, 25 April 1903 – 20 October 1987) was a 20th-century Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.
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Avner Friedman
Avner Friedman (אבנר פרידמן; born November 19, 1932) is Distinguished Professor of Mathematics and Physical Sciences at Ohio State University.
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Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
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École Centrale Paris
École Centrale Paris (ECP, often referred to as École Centrale or Centrale) was a French postgraduate-level institute of research and higher education in engineering and science.
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Baire category theorem
The Baire category theorem (BCT) is an important tool in general topology and functional analysis.
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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Barry Simon
Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and the IBM Professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including the connections to atomic and molecular physics.
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Basis (linear algebra)
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.
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Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra.
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Bounded operator
In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).
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C*-algebra
C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.
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Calculus of variations
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
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Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
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Counting measure
In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be: the number of elements in the subset if the subset has finitely many elements, and ∞ if the subset is infinite.
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Differential equation
A differential equation is a mathematical equation that relates some function with its derivatives.
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Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.
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Dual space
In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
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Ergodic theory
Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.
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Erwin Kreyszig
Erwin O. Kreyszig (January 6, 1922 in Pirna, Germany – December 12, 2008) was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada.
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Essential supremum and essential infimum
In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, i.e., except on a set of measure zero.
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Fourier transform
The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.
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Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.
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Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
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Frigyes Riesz
Frigyes Riesz (Riesz Frigyes,; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators.
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Function space
In mathematics, a function space is a set of functions between two fixed sets.
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Functional (mathematics)
In mathematics, the term functional (as a noun) has at least two meanings.
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George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician.
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Gerald Teschl
Gerald Teschl (born May 12, 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics.
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Haïm Brezis
Haïm Brezis (born 1 June 1944) is a French mathematician who works in functional analysis and partial differential equations.
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Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.
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Hans Hahn (mathematician)
Hans Hahn (27 September 1879 – 24 July 1934) was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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Higher-order function
In mathematics and computer science, a higher-order function (also functional, functional form or functor) is a function that does at least one of the following.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Hugo Steinhaus
Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Jewish-Polish mathematician and educator.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.
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Invariant subspace
In mathematics, an invariant subspace of a linear mapping T: V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.
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Invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself.
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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (ישראל געלפֿאַנד, Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent Soviet mathematician.
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Jacob T. Schwartz
Jacob Theodore "Jack" Schwartz (January 9, 1930 – March 2, 2009) was an American mathematician, computer scientist, and professor of computer science at the New York University Courant Institute of Mathematical Sciences.
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Jacques Hadamard
Jacques Salomon Hadamard ForMemRS (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
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Jean Bourgain
Jean, Baron Bourgain (born 28 February 1954) is a Belgian mathematician.
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John B. Conway
John Bligh Conway (born 1939) is an American mathematician.
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Juliusz Schauder
Juliusz Paweł Schauder (21 September 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and mathematical physics.
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Kōsaku Yosida
was a Japanese mathematician who worked in the field of functional analysis.
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Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.
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Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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Linear form
In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Linear subspace
In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.
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List of functional analysis topics
This is a list of functional analysis topics, by Wikipedia page.
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Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
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Lwów School of Mathematics
The Lwów school of mathematics (lwowska szkoła matematyczna) was a group of Polish mathematicians who worked between the two World Wars in Lwów, Poland (since 1945 Lviv, Ukraine).
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Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
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Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.
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Mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics.
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Maurice René Fréchet
Maurice Fréchet (2 September 1878 – 4 June 1973) was a French mathematician.
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Measurable function
In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.
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Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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Measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.
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Michael C. Reed
Michael (Mike) Charles Reed is the Bishop-MacDermott Professor of Mathematics at Duke University where he has taught since 1977.
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Multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function.
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Nelson Dunford
Nelson James Dunford (December 12, 1906 – September 7, 1986) was an American mathematician, known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators.
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Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense).
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Normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN*.
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Normed vector space
In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.
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Open and closed maps
In topology, an open map is a function between two topological spaces which maps open sets to open sets.
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Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.
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Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
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Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings.
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Operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators.
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Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.
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Paul Lévy (mathematician)
Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions.
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Peter Lax
Peter David Lax (born 1 May 1926) is a Hungarian-born American mathematician working in the areas of pure and applied mathematics.
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Poland
Poland (Polska), officially the Republic of Poland (Rzeczpospolita Polska), is a country located in Central Europe.
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Probability
Probability is the measure of the likelihood that an event will occur.
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Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
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Schauder basis
In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums.
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Separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
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Sergei Fomin
Sergei Vasilyevich Fomin (Серге́й Васи́льевич Фоми́н; 9 December 1917 – 17 August 1975) was a Soviet mathematician who was co-author with Kolmogorov of Introductory real analysis, and co-author with I.M. Gelfand of Calculus of Variations (1963), both books that are widely read in Russian and in English.
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Sobolev
Sobolev (masculine) and Soboleva (feminine) is a popular Russian surname, derived from the word "соболь" (sable).
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Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
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Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.
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Stefan Banach
Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.
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Sublinear function
A sublinear function (or functional, as is more often used in functional analysis), in linear algebra and related areas of mathematics, is a function f: V \rightarrow \mathbf on a vector space V over F, an ordered field (e.g. the real numbers \mathbb), which satisfies \mathbf and any x ∈ V (positive homogeneity), and f(x + y) \le f(x) + f(y) for any x, y ∈ V (subadditivity).
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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Topological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R × R carries the product topology.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topological vector space
In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.
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Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
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Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.
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University of Colorado Colorado Springs
The University of Colorado Colorado Springs (UCCS) is a campus of the University of Colorado system, the state university system of Colorado.
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Up to
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Vito Volterra
Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis.
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Walter Rudin
Walter Rudin (May 2, 1921 – May 20, 2010) was an Austrian-American mathematician and professor of Mathematics at the University of Wisconsin–Madison.
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Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
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References
[1] https://en.wikipedia.org/wiki/Functional_analysis
