Table of Contents
205 relations: Absolute convergence, Absolute value, Affine space, Alexander Grothendieck, Algebra over a field, Anderson–Kadec theorem, Annals of Mathematics, Antilinear map, Approximation property, Baire category theorem, Baire function, Baire space, Balanced set, Ball (mathematics), Banach algebra, Banach manifold, Banach–Alaoglu theorem, Banach–Mazur compactum, Banach–Mazur theorem, Banach–Stone theorem, Barrelled set, Barrelled space, Basis (linear algebra), Bounded operator, Bounded set (topological vector space), C*-algebra, Category theory, Cauchy sequence, Closed graph theorem, Closed set, Closure (topology), Compact operator, Compact space, Comparison of topologies, Complemented subspace, Complete metric space, Complete topological vector space, Complex conjugate, Complex number, Continuous function, Continuous functions on a compact Hausdorff space, Continuous linear operator, Convex combination, Convex function, Convex hull, Convex set, Coproduct, Countable set, David Hilbert, Dense set, ... Expand index (155 more) »
- Normed spaces
Absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.
See Banach space and Absolute convergence
Absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.
See Banach space and Absolute value
Affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.
See Banach space and Affine space
Alexander Grothendieck
Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born mathematician who became the leading figure in the creation of modern algebraic geometry.
See Banach space and Alexander Grothendieck
Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
See Banach space and Algebra over a field
Anderson–Kadec theorem
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. Banach space and Anderson–Kadec theorem are topological vector spaces.
See Banach space and Anderson–Kadec theorem
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Banach space and Annals of Mathematics
Antilinear map
In mathematics, a function f: V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &.
See Banach space and Antilinear map
Approximation property
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. Banach space and approximation property are Banach spaces.
See Banach space and Approximation property
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. Banach space and Baire category theorem are functional analysis.
See Banach space and Baire category theorem
Baire function
In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions.
See Banach space and Baire function
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. Banach space and Baire space are functional analysis.
See Banach space and Baire space
Balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function |\cdot |) is a set S such that a S \subseteq S for all scalars a satisfying |a| \leq 1.
See Banach space and Balanced set
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.
See Banach space and Ball (mathematics)
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. Banach space and Banach algebra are science and technology in Poland.
See Banach space and Banach algebra
Banach manifold
In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Banach space and Banach manifold are Banach spaces.
See Banach space and Banach manifold
Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. Banach space and Banach–Alaoglu theorem are topological vector spaces.
See Banach space and Banach–Alaoglu theorem
Banach–Mazur compactum
In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q(n) of n-dimensional normed spaces. Banach space and Banach–Mazur compactum are functional analysis.
See Banach space and Banach–Mazur compactum
Banach–Mazur theorem
In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. Banach space and Banach–Mazur theorem are functional analysis.
See Banach space and Banach–Mazur theorem
Banach–Stone theorem
In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.
See Banach space and Banach–Stone theorem
Barrelled set
In functional analysis, a subset of a topological vector space (TVS) is called a barrel or a barrelled set if it is closed convex balanced and absorbing. Banach space and barrelled set are topological vector spaces.
See Banach space and Barrelled set
Barrelled space
In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. Banach space and barrelled space are topological vector spaces.
See Banach space and Barrelled space
Basis (linear algebra)
In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.
See Banach space and Basis (linear algebra)
Bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L: X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \|Lx\|_Y \leq M \|x\|_X.
See Banach space and Bounded operator
Bounded set (topological vector space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. Banach space and bounded set (topological vector space) are topological vector spaces.
See Banach space and Bounded set (topological vector space)
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. Banach space and C*-algebra are functional analysis.
See Banach space and C*-algebra
Category theory
Category theory is a general theory of mathematical structures and their relations.
See Banach space and Category theory
Cauchy sequence
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
See Banach space and Cauchy sequence
Closed graph theorem
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
See Banach space and Closed graph theorem
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
See Banach space and Closed set
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of.
See Banach space and Closure (topology)
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y).
See Banach space and Compact operator
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.
See Banach space and Compact space
Comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.
See Banach space and Comparison of topologies
Complemented subspace
In the branch of mathematics called functional analysis, a M of V has an algebraic complement: another vector subspace N such that V. Banach space and Complemented subspace are functional analysis.
See Banach space and Complemented subspace
Complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in.
See Banach space and Complete metric space
Complete topological vector space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards which they all get closer. Banach space and complete topological vector space are functional analysis and topological vector spaces.
See Banach space and Complete topological vector space
Complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
See Banach space and Complex conjugate
Complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.
See Banach space and Complex number
Continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.
See Banach space and Continuous function
Continuous functions on a compact Hausdorff space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X with values in the real or complex numbers. Banach space and continuous functions on a compact Hausdorff space are Banach spaces and functional analysis.
See Banach space and Continuous functions on a compact Hausdorff space
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. Banach space and continuous linear operator are functional analysis.
See Banach space and Continuous linear operator
Convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.
See Banach space and Convex combination
Convex function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points.
See Banach space and Convex function
Convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it.
See Banach space and Convex hull
Convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them.
See Banach space and Convex set
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.
See Banach space and Coproduct
Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.
See Banach space and Countable set
David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of his time.
See Banach space and David Hilbert
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
See Banach space and Dense set
Derived set (mathematics)
In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by S'.
See Banach space and Derived set (mathematics)
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.
See Banach space and Dimension (vector space)
Dirac measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not.
See Banach space and Dirac measure
Directional derivative
A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point.
See Banach space and Directional derivative
Disk algebra
In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions (where D is the open unit disk in the complex plane \mathbb) that extend to a continuous function on the closure of D. That is, where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. Banach space and disk algebra are functional analysis.
See Banach space and Disk algebra
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Banach space and distribution (mathematics) are functional analysis.
See Banach space and Distribution (mathematics)
Dixmier–Ng theorem
In functional analysis, the Dixmier–Ng theorem is a characterization of when a normed space is in fact a dual Banach space.
See Banach space and Dixmier–Ng theorem
Dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Banach space and dual norm are functional analysis.
See Banach space and Dual norm
Dvoretzky's theorem
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. Banach space and Dvoretzky's theorem are Banach spaces.
See Banach space and Dvoretzky's theorem
Eberlein–Šmulian theorem
In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in a Banach space. Banach space and Eberlein–Šmulian theorem are Banach spaces.
See Banach space and Eberlein–Šmulian theorem
Eduard Helly
Eduard Helly (June 1, 1884 in Vienna – 28 November 1943 in Chicago) was a mathematician after whom Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem were named.
See Banach space and Eduard Helly
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
See Banach space and Embedding
Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
See Banach space and Equivalence relation
Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an American mathematician known for his work on analysis.
See Banach space and Errett Bishop
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted \textbf^2 or \mathbb^2.
See Banach space and Euclidean plane
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
See Banach space and Euclidean topology
Extreme point
In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S that does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or corner point of S. Banach space and extreme point are functional analysis.
See Banach space and Extreme point
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d: X \times X \to \R such that. Banach space and f-space are topological vector spaces.
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.
See Banach space and Field (mathematics)
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output.
See Banach space and Forgetful functor
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Banach space and Fréchet derivative are Banach spaces.
See Banach space and Fréchet derivative
Fréchet manifold
In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.
See Banach space and Fréchet manifold
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. Banach space and Fréchet space are topological vector spaces.
See Banach space and Fréchet space
Fréchet–Urysohn space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the sequential closure of S in X. Fréchet–Urysohn spaces are a special type of sequential space.
See Banach space and Fréchet–Urysohn space
Frigyes Riesz
Frigyes Riesz (Riesz Frigyes,, sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators.
See Banach space and Frigyes Riesz
Function space
In mathematics, a function space is a set of functions between two fixed sets.
See Banach space and Function space
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 (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
See Banach space and Functional analysis
Functor
In mathematics, specifically category theory, a functor is a mapping between categories.
Gateaux derivative
In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Banach space and Gateaux derivative are topological vector spaces.
See Banach space and Gateaux derivative
Gelfand representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things. Banach space and Gelfand representation are functional analysis.
See Banach space and Gelfand representation
Gelfand–Mazur theorem
In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states that a Banach algebra with unit over the complex numbers in which every nonzero element is invertible is isometrically isomorphic to the complex numbers, i. e., the only complex Banach algebra that is a division algebra is the complex numbers C.
See Banach space and Gelfand–Mazur theorem
Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space.
See Banach space and Gelfand–Naimark theorem
Gilles Pisier
Gilles I. Pisier (born 18 November 1950) is a professor of mathematics at the Pierre and Marie Curie University and a distinguished professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University.
See Banach space and Gilles Pisier
Goldstine theorem
In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, is stated as follows: The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, c0 space c_0, and its bi-dual space Lp space \ell^. Banach space and Goldstine theorem are Banach spaces.
See Banach space and Goldstine theorem
Ground field
In mathematics, a ground field is a field K fixed at the beginning of the discussion.
See Banach space and Ground field
Haar wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis.
See Banach space and Haar wavelet
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis. Banach space and Hahn–Banach theorem are topological vector spaces.
See Banach space and Hahn–Banach theorem
Hans Hahn (mathematician)
Hans Hahn (27 September 1879 – 24 July 1934) was an Austrian mathematician and philosopher who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.
See Banach space and Hans Hahn (mathematician)
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane.
See Banach space and Hardy space
Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.
See Banach space and Harmonic analysis
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each that are disjoint from each other.
See Banach space and Hausdorff space
Hermitian function
In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: (where the ^* indicates the complex conjugate) for all x in the domain of f. In physics, this property is referred to as PT symmetry.
See Banach space and Hermitian function
Hilbert manifold
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces.
See Banach space and Hilbert manifold
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Banach space and Hilbert space are functional analysis.
See Banach space and Hilbert space
Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.
See Banach space and Holomorphic function
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
See Banach space and Homeomorphism
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.
See Banach space and Hyperplane
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B: \iota: A\rightarrow B, \qquad \iota(x).
See Banach space and Inclusion map
Infinite-dimensional optimization
In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Banach space and Infinite-dimensional optimization are functional analysis.
See Banach space and Infinite-dimensional optimization
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.
See Banach space and Injective function
Inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. Banach space and inner product space are normed spaces.
See Banach space and Inner product space
Interior (topology)
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in.
See Banach space and Interior (topology)
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, for all in the domain of.
See Banach space and Involution (mathematics)
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (ישראל געלפֿאַנד, Изра́иль Моисе́евич Гельфа́нд, Ізраїль Мойсейович Гельфанд; – 5 October 2009) was a prominent Soviet-American mathematician.
See Banach space and Israel Gelfand
Israel Journal of Mathematics
Israel Journal of Mathematics is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press).
See Banach space and Israel Journal of Mathematics
James's theorem
In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X is reflexive if and only if every continuous linear functional's norm on X attains its supremum on the closed unit ball in X. A stronger version of the theorem states that a weakly closed subset C of a Banach space X is weakly compact if and only if the dual norm each continuous linear functional on X attains a maximum on C.
See Banach space and James's theorem
Kolmogorov's normability criterion
In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be; that is, for the existence of a norm on the space that generates the given topology.
See Banach space and Kolmogorov's normability criterion
L-semi-inner product
In mathematics, there are two different notions of semi-inner-product. Banach space and l-semi-inner product are functional analysis.
See Banach space and L-semi-inner product
LF-space
In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. Banach space and lF-space are topological vector spaces.
Limit of a sequence
As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1.
See Banach space and Limit of a sequence
Linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers). Banach space and linear form are functional analysis.
See Banach space and Linear form
Linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.
See Banach space and Linear map
Linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted, pp.
See Banach space and Linear span
Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspaceThe term linear subspace is sometimes used for referring to flats and affine subspaces. Banach space and linear subspace are functional analysis.
See Banach space and Linear subspace
Local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
See Banach space and Local homeomorphism
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
See Banach space and Locally compact space
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. Banach space and locally convex topological vector space are functional analysis and topological vector spaces.
See Banach space and Locally convex topological vector space
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces. Banach space and Lp space are Banach spaces and normed spaces.
Mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
See Banach space and Mathematical analysis
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Banach space and Mathematics
Mazur–Ulam theorem
In mathematics, the Mazur–Ulam theorem states that if V and W are normed spaces over R and the mapping is a surjective isometry, then f is affine. Banach space and Mazur–Ulam theorem are normed spaces.
See Banach space and Mazur–Ulam theorem
Metric map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance.
See Banach space and Metric map
Metric space
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.
See Banach space and Metric space
Metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
See Banach space and Metrizable space
Metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). Banach space and metrizable topological vector space are topological vector spaces.
See Banach space and Metrizable topological vector space
Microbundle
In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964.
See Banach space and Microbundle
Milman–Pettis theorem
In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. Banach space and Milman–Pettis theorem are Banach spaces.
See Banach space and Milman–Pettis theorem
Montel space
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Banach space and Montel space are functional analysis and topological vector spaces.
See Banach space and Montel space
Morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.
Natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.
See Banach space and Natural number
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
See Banach space and Neighbourhood (mathematics)
Neighbourhood system
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
See Banach space and Neighbourhood system
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set.
See Banach space and Net (mathematics)
Nicole Tomczak-Jaegermann
Nicole Tomczak-Jaegermann FRSC (8 June 1945 – 17 June 2022) was a Polish-Canadian mathematician, a professor of mathematics at the University of Alberta, and the holder of the Canada Research Chair in Geometric Analysis.
See Banach space and Nicole Tomczak-Jaegermann
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. Banach space and norm (mathematics) are functional analysis.
See Banach space and Norm (mathematics)
Normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. Banach space and normed vector space are normed spaces.
See Banach space and Normed vector space
Open and closed maps
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
See Banach space and Open and closed maps
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
See Banach space and Open mapping theorem (functional analysis)
Open set
In mathematics, an open set is a generalization of an open interval in the real line.
Operator
Operator may refer to.
Operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its. Banach space and operator norm are functional analysis.
See Banach space and Operator norm
Order topology
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set.
See Banach space and Order topology
Ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
See Banach space and Ordinal number
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. Banach space and orthonormal basis are functional analysis.
See Banach space and Orthonormal basis
Parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry.
See Banach space and Parallelogram law
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.
See Banach space and Parseval's theorem
Partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
See Banach space and Partial differential equation
Per Enflo
Per H. Enflo (born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental.
See Banach space and Per Enflo
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.
See Banach space and Pointwise convergence
Polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Banach space and polarization identity are functional analysis.
See Banach space and Polarization identity
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies measure properties such as countable additivity.
See Banach space and Probability measure
Product (category theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
See Banach space and Product (category theory)
Product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology.
See Banach space and Product topology
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P. Banach space and projection (linear algebra) are functional analysis.
See Banach space and Projection (linear algebra)
Quasi-derivative
In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. Banach space and quasi-derivative are Banach spaces.
See Banach space and Quasi-derivative
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets.
See Banach space and Radon measure
Range of a function
In mathematics, the range of a function may refer to either of two closely related concepts.
See Banach space and Range of a function
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Banach space and Real number
Reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is a homeomorphism (or equivalently, a TVS isomorphism). Banach space and reflexive space are Banach spaces.
See Banach space and Reflexive space
René Maurice Fréchet
René Maurice Fréchet (2 September 1878 – 4 June 1973) was a French mathematician.
See Banach space and René Maurice Fréchet
Riesz representation theorem
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space.
See Banach space and Riesz representation theorem
Robert Phelps
Robert Ralph Phelps (March 22, 1926 – January 4, 2013) was an American mathematician who was known for his contributions to analysis, particularly to functional analysis and measure theory.
See Banach space and Robert Phelps
Schur's property
In mathematics, Schur's property, named after Issai Schur, is the property of normed spaces that is satisfied precisely if weak convergence of sequences entails convergence in norm. Banach space and Schur's property are functional analysis.
See Banach space and Schur's property
Seminorm
In mathematics, particularly in functional analysis, a seminorm is a norm that need not be positive definite.
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.
See Banach space and Separable space
Sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Banach space and sequence space are functional analysis.
See Banach space and Sequence space
Sequentially compact space
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice).
See Banach space and Sequentially compact space
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function.
See Banach space and Smooth structure
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order.
See Banach space and Sobolev space
Spaces of test functions and distributions
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Banach space and spaces of test functions and distributions are functional analysis and topological vector spaces.
See Banach space and Spaces of test functions and distributions
Spectrum of a C*-algebra
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and which is invariant under all operators π(x) with x ∈ A.
See Banach space and Spectrum of a C*-algebra
Square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Banach space and square-integrable function are functional analysis.
See Banach space and Square-integrable function
Stefan Banach
Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians.
See Banach space and Stefan Banach
Stefan Mazurkiewicz
Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability.
See Banach space and Stefan Mazurkiewicz
Strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of X, where this topology is denoted by b\left(X^, X\right) or \beta\left(X^, X\right). Banach space and strong dual space are functional analysis.
See Banach space and Strong dual space
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.
See Banach space and Subadditivity
Sublinear function
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. Banach space and sublinear function are functional analysis.
See Banach space and Sublinear function
Surjective function
In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that.
See Banach space and Surjective function
Tensor (intrinsic definition)
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept.
See Banach space and Tensor (intrinsic definition)
Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.
See Banach space and Tensor product
Timothy Gowers
Sir William Timothy Gowers, (born 20 November 1963) is a British mathematician.
See Banach space and Timothy Gowers
Topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
See Banach space and Topological group
Topological manifold
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space.
See Banach space and Topological manifold
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.
See Banach space and Topological space
Topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. Banach space and topological vector space are topological vector spaces.
See Banach space and Topological vector space
Topologies on spaces of linear maps
In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Banach space and topologies on spaces of linear maps are functional analysis and topological vector spaces.
See Banach space and Topologies on spaces of linear maps
Topology
Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Total derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments.
See Banach space and Total derivative
Totally bounded space
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. Banach space and Totally bounded space are functional analysis.
See Banach space and Totally bounded space
Translational symmetry
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation).
See Banach space and Translational symmetry
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology.
See Banach space and Tychonoff's theorem
Type and cotype of a Banach space
In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure, how far a Banach space from a Hilbert space is. Banach space and type and cotype of a Banach space are Banach spaces and functional analysis.
See Banach space and Type and cotype of a Banach space
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Banach space and uniform boundedness principle are functional analysis.
See Banach space and Uniform boundedness principle
Uniform continuity
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want.
See Banach space and Uniform continuity
Uniform space
In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence.
See Banach space and Uniform space
Uniformly convex space
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. Banach space and uniformly convex space are Banach spaces.
See Banach space and Uniformly convex space
Unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1.
See Banach space and Unit circle
Unit sphere
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. Banach space and unit sphere are functional analysis.
See Banach space and Unit sphere
Universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions.
See Banach space and Universal property
Up to
Two mathematical objects and are called "equal up to an equivalence relation ".
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.
See Banach space and Vector space
Wacław Sierpiński
Wacław Franciszek Sierpiński (14 March 1882 – 21 October 1969) was a Polish mathematician.
See Banach space and Wacław Sierpiński
Weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.
See Banach space and Weak topology
Wiener algebra
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by, is the space of absolutely convergent Fourier series.
See Banach space and Wiener algebra
See also
Normed spaces
- Banach space
- Banach spaces
- C space
- Cocompact embedding
- Fichera's existence principle
- Holmes–Thompson volume
- Inner product space
- Krein–Smulian theorem
- L-infinity
- Lp space
- Lp spaces
- Mazur–Ulam theorem
- Minkowski distance
- Minkowski's first inequality for convex bodies
- Normed vector space
- Riesz's lemma
- Strictly convex space
- Uniform norm
References
Also known as Banach Spaces, Banach function space, Banach norm, Complete norm, Complete normed space, Complete normed vector space, Dual banach space, Isomorphic normed spaces, Linear Algebra/Banach Spaces.
, Derived set (mathematics), Dimension (vector space), Dirac measure, Directional derivative, Disk algebra, Distribution (mathematics), Dixmier–Ng theorem, Dual norm, Dvoretzky's theorem, Eberlein–Šmulian theorem, Eduard Helly, Embedding, Equivalence relation, Errett Bishop, Euclidean plane, Euclidean topology, Extreme point, F-space, Field (mathematics), Forgetful functor, Fréchet derivative, Fréchet manifold, Fréchet space, Fréchet–Urysohn space, Frigyes Riesz, Function space, Functional analysis, Functor, Gateaux derivative, Gelfand representation, Gelfand–Mazur theorem, Gelfand–Naimark theorem, Gilles Pisier, Goldstine theorem, Ground field, Haar wavelet, Hahn–Banach theorem, Hans Hahn (mathematician), Hardy space, Harmonic analysis, Hausdorff space, Hermitian function, Hilbert manifold, Hilbert space, Holomorphic function, Homeomorphism, Hyperplane, Inclusion map, Infinite-dimensional optimization, Injective function, Inner product space, Interior (topology), Involution (mathematics), Isometry, Israel Gelfand, Israel Journal of Mathematics, James's theorem, Kolmogorov's normability criterion, L-semi-inner product, LF-space, Limit of a sequence, Linear form, Linear map, Linear span, Linear subspace, Local homeomorphism, Locally compact space, Locally convex topological vector space, Lp space, Mathematical analysis, Mathematics, Mazur–Ulam theorem, Metric map, Metric space, Metrizable space, Metrizable topological vector space, Microbundle, Milman–Pettis theorem, Montel space, Morphism, Natural number, Neighbourhood (mathematics), Neighbourhood system, Net (mathematics), Nicole Tomczak-Jaegermann, Norm (mathematics), Normed vector space, Open and closed maps, Open mapping theorem (functional analysis), Open set, Operator, Operator norm, Order topology, Ordinal number, Orthonormal basis, Parallelogram law, Parseval's theorem, Partial differential equation, Per Enflo, Pointwise convergence, Polarization identity, Probability measure, Product (category theory), Product topology, Projection (linear algebra), Quasi-derivative, Radon measure, Range of a function, Real number, Reflexive space, René Maurice Fréchet, Riesz representation theorem, Robert Phelps, Schur's property, Seminorm, Separable space, Sequence space, Sequentially compact space, Smooth structure, Sobolev space, Spaces of test functions and distributions, Spectrum of a C*-algebra, Square-integrable function, Stefan Banach, Stefan Mazurkiewicz, Strong dual space, Subadditivity, Sublinear function, Surjective function, Tensor (intrinsic definition), Tensor product, Timothy Gowers, Topological group, Topological manifold, Topological space, Topological vector space, Topologies on spaces of linear maps, Topology, Total derivative, Totally bounded space, Translational symmetry, Tychonoff's theorem, Type and cotype of a Banach space, Uniform boundedness principle, Uniform continuity, Uniform space, Uniformly convex space, Unit circle, Unit sphere, Universal property, Up to, Vector space, Wacław Sierpiński, Weak topology, Wiener algebra.