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42 relations: Axiom of choice, Banach space, Banach–Tarski paradox, Boolean prime ideal theorem, Bounded operator, Closure (topology), Continuous function, Convex function, Convex geometry, Dover Publications, Dual space, Eduard Helly, Farkas' lemma, Fichera's existence principle, Function (mathematics), Functional analysis, Hans Hahn (mathematician), Homogeneous function, Hyperplane separation theorem, Injective object, Inventiones Mathematicae, König's lemma, Linear form, Linear subspace, Locally convex topological vector space, M. Riesz extension theorem, Marcel Riesz, Mathematical optimization, Mathematics, Minkowski functional, Mizar system, Norm (mathematics), Normed vector space, Reverse mathematics, Second-order arithmetic, Separable space, Stefan Banach, Subadditivity, Sublinear function, Topological vector space, Vector space, Zorn's lemma.
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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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.
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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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Closure (topology)
In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
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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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Convex function
In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.
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Convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.
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Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.
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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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Eduard Helly
Eduard Helly (June 1, 1884, Vienna – 1943, Chicago) was a mathematician after whom Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem were named.
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Farkas' lemma
Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics.
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Fichera's existence principle
In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
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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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Homogeneous function
In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
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Hyperplane separation theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.
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Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module.
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Inventiones Mathematicae
Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.
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König's lemma
König's lemma or Kőnig's infinity lemma is a theorem in graph theory due to.
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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 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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Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces.
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M. Riesz extension theorem
The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.
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Marcel Riesz
Marcel Riesz (Riesz Marcell; 16 November 1886 – 4 September 1969) was a Hungarian-born mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras.
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Mathematical optimization
In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional is a function that recovers a notion of distance on a linear space.
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Mizar system
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in the proof of new theorems.
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Norm (mathematics)
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.
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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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Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.
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Second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets.
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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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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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Subadditivity
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element.
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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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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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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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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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Hahn Banach Theorem, Hahn Banach theorem, Hahn banach extension theorem, Hahn banach theorem, Hahn-Banach, Hahn-Banach theorem, Hahn–Banach separation theorem.
References
[1] https://en.wikipedia.org/wiki/Hahn–Banach_theorem
