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51 relations: Additive map, Annals of Mathematics, Axiom of choice, Axiom of dependent choice, Banach space, Basis (linear algebra), Bounded operator, Closed graph theorem, Commensurability (mathematics), Complete metric space, Constructivism (mathematics), Continuous function, Densely defined operator, Derivative, Dimension (vector space), Dual space, F-space, Function (mathematics), Group (mathematics), Hahn–Banach theorem, Infimum and supremum, Lebesgue measure, Linear approximation, Linear form, Linear independence, Linear map, Lp space, Mathematics, Measurable function, Minkowski functional, Normed vector space, Nowhere continuous function, Polynomial, Property of Baire, Quasinorm, Rational number, Real number, Robert M. Solovay, Sequence, Set theory, Smoothness, Stone–Weierstrass theorem, Structure (mathematical logic), Topological space, Topological vector space, Triangle inequality, Unbounded operator, Uniform norm, Vector space, Vitali set, ..., Zermelo–Fraenkel set theory. Expand index (1 more) »
Additive map
In algebra an additive map, Z-linear map or additive function is a function that preserves the addition operation: for every pair of elements and in the domain.
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Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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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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Axiom of dependent choice
In mathematics, the axiom of dependent choice, denoted by \mathsf, is a weak form of the axiom of choice (\mathsf) that is still sufficient to develop most of real 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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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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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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Closed graph theorem
In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs.
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Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio is a rational number; otherwise a and b are called incommensurable.
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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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Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.
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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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Densely defined operator
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function.
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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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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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F-space
In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d: V × V → R so that.
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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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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.
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Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
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Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.
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Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
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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 independence
In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.
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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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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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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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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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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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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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Nowhere continuous function
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Property of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such that A \bigtriangleup U is meager (where \bigtriangleup denotes the symmetric difference).
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Quasinorm
In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by for some K > 0.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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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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Robert M. Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function.
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Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.
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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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Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
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Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
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Uniform norm
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.
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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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Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali.
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Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
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A linear functional which is not continuous, A linear map which is not continuous, Discontinuous linear function, Discontinuous linear functional, Discontinuous linear operator, General existence theorem of discontinuous maps, Linear discontinuous map, Linear operator which is not continuous, Non-continuous linear functional.
References
[1] https://en.wikipedia.org/wiki/Discontinuous_linear_map
