Faster access than browser!
158 relations: Absolute continuity, Absolute convergence, Absolute value, Affine space, Alexander Grothendieck, Algebra over a field, Annals of Mathematics, Antilinear map, Approximation property, Ba space, Baire category theorem, Baire function, Ball (mathematics), Banach algebra, Banach–Alaoglu theorem, Banach–Mazur compactum, Banach–Mazur theorem, Banach–Stone theorem, Basis (linear algebra), Bounded operator, Bounded variation, Bs space, C space, C*-algebra, Cauchy sequence, Closed graph theorem, Closed set, Closure (topology), Compact operator, Compact space, Comparison of topologies, Complete metric space, Complex conjugate, Complex number, Continuous function, Continuous functions on a compact Hausdorff space, Convex function, Convex set, Countable set, David Hilbert, Dense set, Derived set (mathematics), Differentiable function, Dirac measure, Directional derivative, Disk algebra, Distortion problem, Distribution (mathematics), Dual space, Dvoretzky's theorem, ..., Eberlein–Šmulian theorem, Eduard Helly, Errett Bishop, Euclidean space, Extreme point, F-space, Field (mathematics), Fréchet derivative, Fréchet space, Frigyes Riesz, Function space, Functional analysis, Gâteaux 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 space, Holomorphic function, Homeomorphism, Hyperplane, Infinite-dimensional optimization, Injective function, Inner product space, Interior (topology), Interpolation space, Involution (mathematics), Isometry, Israel Gelfand, James's theorem, LF-space, Limit of a sequence, Linear form, Linear map, Linear span, Linear subspace, Locally convex topological vector space, Lp space, Mathematical analysis, Mathematics, Maurice René Fréchet, Measure (mathematics), Metric (mathematics), Metric space, Metrization theorem, Milman–Pettis theorem, Natural number, Net (mathematics), Nicole Tomczak-Jaegermann, Norm (mathematics), Normed vector space, Open mapping theorem (functional analysis), Operator, Operator norm, Order topology, Ordinal number, Parallelogram law, Parseval's theorem, Partial differential equation, Per Enflo, Pointwise convergence, Polarization identity, Probability measure, Projection (linear algebra), Quasi-derivative, Radon measure, Range (mathematics), Real number, Reflexive space, Riesz representation theorem, Robert Phelps, Schur's property, Separable space, Sequence space, Series (mathematics), Sigma-algebra, Smith space, Sobolev space, Space (mathematics), Spectrum of a C*-algebra, Stefan Banach, Stefan Mazurkiewicz, Sublinear function, Surjective function, Taylor's theorem, Tensor product, Timothy Gowers, Topological vector space, Total variation, Tychonoff's theorem, Uniform boundedness principle, Uniform space, Uniformly convex space, Unit sphere, Universal property, Vector space, Wacław Sierpiński, Weak topology, Wiener algebra. Expand index (108 more) »
Absolute continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.
New!!: Banach space and Absolute continuity · See more »
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.
New!!: Banach space and Absolute convergence · See 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.
New!!: Banach space and Absolute value · See more »
Affine space
In mathematics, an affine space is a geometric structure that generalizes 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.
New!!: Banach space and Affine space · See more »
Alexander Grothendieck
Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry.
New!!: Banach space and Alexander Grothendieck · See more »
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.
New!!: Banach space and Algebra over a field · See more »
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
New!!: Banach space and Annals of Mathematics · See more »
Antilinear map
In mathematics, a mapping f:V\to W from a complex vector space to another is said to be antilinear (or conjugate-linear) if for all a, \, b \, \in \mathbb and all x, \, y \, \in V, where \bar and \bar are the complex conjugates of a and b respectively.
New!!: Banach space and Antilinear map · See more »
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.
New!!: Banach space and Approximation property · See more »
Ba space
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma.
New!!: Banach space and Ba space · See more »
Baire category theorem
The Baire category theorem (BCT) is an important tool in general topology and functional analysis.
New!!: Banach space and Baire category theorem · See more »
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.
New!!: Banach space and Baire function · See more »
Ball (mathematics)
In mathematics, a ball is the space bounded by a sphere.
New!!: Banach space and Ball (mathematics) · See more »
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, i.e. a normed space and complete in the metric induced by the norm.
New!!: Banach space and Banach algebra · See more »
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.
New!!: Banach space and Banach–Alaoglu theorem · See more »
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.
New!!: Banach space and Banach–Mazur compactum · See more »
Banach–Mazur theorem
In mathematics, the Banach–Mazur theorem is a theorem of functional analysis.
New!!: Banach space and Banach–Mazur theorem · See more »
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.
New!!: Banach space and Banach–Stone theorem · See more »
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.
New!!: Banach space and Basis (linear algebra) · See more »
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).
New!!: Banach space and Bounded operator · See more »
Bounded variation
In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense.
New!!: Banach space and Bounded variation · See more »
Bs space
In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real or complex numbers such that is finite.
New!!: Banach space and Bs space · See more »
C space
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences (xn) of real numbers or complex numbers.
New!!: Banach space and C space · See more »
C*-algebra
C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.
New!!: Banach space and C*-algebra · See more »
Cauchy sequence
In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
New!!: Banach space and Cauchy sequence · See more »
Closed graph theorem
In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs.
New!!: Banach space and Closed graph theorem · See more »
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
New!!: Banach space and Closed set · See more »
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.
New!!: Banach space and Closure (topology) · See more »
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.
New!!: Banach space and Compact operator · See more »
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).
New!!: Banach space and Compact space · See more »
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.
New!!: Banach space and Comparison of topologies · See more »
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).
New!!: Banach space and Complete metric space · See more »
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.
New!!: Banach space and Complex conjugate · See more »
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.
New!!: Banach space and Complex number · See more »
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.
New!!: Banach space and Continuous function · See more »
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 with values in the real or complex numbers.
New!!: Banach space and Continuous functions on a compact Hausdorff space · See more »
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.
New!!: Banach space and Convex function · See more »
Convex set
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.
New!!: Banach space and Convex set · See more »
Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
New!!: Banach space and Countable set · See more »
David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
New!!: Banach space and David Hilbert · See more »
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
New!!: Banach space and Dense set · See more »
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'.
New!!: Banach space and Derived set (mathematics) · See more »
Differentiable function
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
New!!: Banach space and Differentiable function · See more »
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.
New!!: Banach space and Dirac measure · See more »
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
New!!: Banach space and Directional derivative · See more »
Disk algebra
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 C, f extends to a continuous function on the closure of D. That is, where H^\infty(\mathbf) denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space).
New!!: Banach space and Disk algebra · See more »
Distortion problem
In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm.
New!!: Banach space and Distortion problem · See more »
Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.
New!!: Banach space and Distribution (mathematics) · See more »
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.
New!!: Banach space and Dual space · See more »
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.
New!!: Banach space and Dvoretzky's theorem · See more »
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.
New!!: Banach space and Eberlein–Šmulian theorem · See more »
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.
New!!: Banach space and Eduard Helly · See more »
Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an American mathematician known for his work on analysis.
New!!: Banach space and Errett Bishop · See more »
Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
New!!: Banach space and Euclidean space · See more »
Extreme point
In mathematics, an extreme point of a convex set S in a real vector space is a point in S which does not lie in any open line segment joining two points of S. Intuitively, an extreme point is a "vertex" of S.
New!!: Banach space and Extreme point · See more »
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.
New!!: Banach space and F-space · See more »
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
New!!: Banach space and Field (mathematics) · See more »
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces.
New!!: Banach space and Fréchet derivative · See more »
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
New!!: Banach space and Fréchet space · See more »
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.
New!!: Banach space and Frigyes Riesz · See more »
Function space
In mathematics, a function space is a set of functions between two fixed sets.
New!!: Banach space and Function space · See more »
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.
New!!: Banach space and Functional analysis · See more »
Gâteaux derivative
In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus.
New!!: Banach space and Gâteaux derivative · See more »
Gelfand representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings.
New!!: Banach space and Gelfand representation · See more »
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. The theorem follows from the fact that the spectrum of any element of a complex Banach algebra is nonempty: for every element a of a complex Banach algebra A there is some complex number λ such that λ1 − a is not invertible.
New!!: Banach space and Gelfand–Mazur theorem · See more »
Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space.
New!!: Banach space and Gelfand–Naimark theorem · See more »
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.
New!!: Banach space and Gilles Pisier · See more »
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, ''c''0, and its bi-dual space ℓ∞.
New!!: Banach space and Goldstine theorem · See more »
Ground field
In mathematics, a ground field is a field K fixed at the beginning of the discussion.
New!!: Banach space and Ground field · See more »
Haar wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis.
New!!: Banach space and Haar wavelet · See more »
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.
New!!: Banach space and Hahn–Banach theorem · See more »
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.
New!!: Banach space and Hans Hahn (mathematician) · See more »
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.
New!!: Banach space and Hardy space · See more »
Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).
New!!: Banach space and Harmonic analysis · See more »
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.
New!!: Banach space and Hausdorff space · See more »
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. This definition extends also to functions of two or more variables, e.g., in the case that f is a function of two variables it is Hermitian if for all pairs (x_1, x_2) in the domain of f. From this definition it follows immediately that: f is a Hermitian function if and only if.
New!!: Banach space and Hermitian function · See more »
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
New!!: Banach space and Hilbert space · See more »
Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
New!!: Banach space and Holomorphic function · See more »
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
New!!: Banach space and Homeomorphism · See more »
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.
New!!: Banach space and Hyperplane · See more »
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.
New!!: Banach space and Infinite-dimensional optimization · See more »
Injective function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
New!!: Banach space and Injective function · See more »
Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
New!!: Banach space and Inner product space · See more »
Interior (topology)
In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.
New!!: Banach space and Interior (topology) · See more »
Interpolation space
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces.
New!!: Banach space and Interpolation space · See more »
Involution (mathematics)
In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.
New!!: Banach space and Involution (mathematics) · See more »
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
New!!: Banach space and Isometry · See more »
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.
New!!: Banach space and Israel Gelfand · See more »
James's theorem
In mathematics, particularly functional analysis, James's theorem, named for Robert C. James, states that a Banach space B is reflexive if and only if every continuous linear functional on B attains its supremum on the closed unit ball in B. A stronger version of the theorem states that a weakly closed subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C. The hypothesis of completeness in the theorem cannot be dropped.
New!!: Banach space and James's theorem · See more »
LF-space
In mathematics, an LF-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system (V_n, i_) of Fréchet spaces.
New!!: Banach space and LF-space · See more »
Limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.
New!!: Banach space and Limit of a sequence · See more »
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.
New!!: Banach space and Linear form · See more »
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.
New!!: Banach space and Linear map · See more »
Linear span
In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.
New!!: Banach space and Linear span · See more »
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.
New!!: Banach space and Linear subspace · See more »
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.
New!!: Banach space and Locally convex topological vector space · See more »
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
New!!: Banach space and Lp space · See more »
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.
New!!: Banach space and Mathematical analysis · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Banach space and Mathematics · See more »
Maurice René Fréchet
Maurice Fréchet (2 September 1878 – 4 June 1973) was a French mathematician.
New!!: Banach space and Maurice René Fréchet · See more »
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.
New!!: Banach space and Measure (mathematics) · See more »
Metric (mathematics)
In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.
New!!: Banach space and Metric (mathematics) · See more »
Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
New!!: Banach space and Metric space · See more »
Metrization theorem
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
New!!: Banach space and Metrization theorem · See more »
Milman–Pettis theorem
In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive.
New!!: Banach space and Milman–Pettis theorem · See more »
Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
New!!: Banach space and Natural number · See more »
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.
New!!: Banach space and Net (mathematics) · See more »
Nicole Tomczak-Jaegermann
Nicole Tomczak-Jaegermann FRSC is a Polish Canadian mathematician, a professor of mathematics at the University of Alberta, and the holder of the Canada Research Chair in Geometric Analysis.
New!!: Banach space and Nicole Tomczak-Jaegermann · See more »
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.
New!!: Banach space and Norm (mathematics) · See more »
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.
New!!: Banach space and Normed vector space · See more »
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.
New!!: Banach space and Open mapping theorem (functional analysis) · See more »
Operator
Operator may refer to.
New!!: Banach space and Operator · See more »
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators.
New!!: Banach space and Operator norm · See more »
Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set.
New!!: Banach space and Order topology · See more »
Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
New!!: Banach space and Ordinal number · See more »
Parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry.
New!!: Banach space and Parallelogram law · See more »
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.
New!!: Banach space and Parseval's theorem · See more »
Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
New!!: Banach space and Partial differential equation · See more »
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.
New!!: Banach space and Per Enflo · See more »
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.
New!!: Banach space and Pointwise convergence · See more »
Polarization identity
In 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.
New!!: Banach space and Polarization identity · See more »
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.
New!!: Banach space and Probability measure · See more »
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
New!!: Banach space and Projection (linear algebra) · See more »
Quasi-derivative
In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces.
New!!: Banach space and Quasi-derivative · See more »
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 X that is locally finite and inner regular.
New!!: Banach space and Radon measure · See more »
Range (mathematics)
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.
New!!: Banach space and Range (mathematics) · See more »
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
New!!: Banach space and Real number · See more »
Reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space.
New!!: Banach space and Reflexive space · See more »
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem.
New!!: Banach space and Riesz representation theorem · See more »
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.
New!!: Banach space and Robert Phelps · See more »
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.
New!!: Banach space and Schur's property · See more »
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.
New!!: Banach space and Separable space · See more »
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.
New!!: Banach space and Sequence space · See more »
Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
New!!: Banach space and Series (mathematics) · See more »
Sigma-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.
New!!: Banach space and Sigma-algebra · See more »
Smith space
In functional analysis and related areas of mathematics, Smith space is a complete compactly generated locally convex space X having a compact set K which absorbs every other compact set T\subseteq X (i.e. T\subseteq\lambda\cdot K for some \lambda>0).
New!!: Banach space and Smith space · See more »
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 itself and its derivatives up to a given order.
New!!: Banach space and Sobolev space · See more »
Space (mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure.
New!!: Banach space and Space (mathematics) · See more »
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. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces.
New!!: Banach space and Spectrum of a C*-algebra · See more »
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.
New!!: Banach space and Stefan Banach · See more »
Stefan Mazurkiewicz
Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability.
New!!: Banach space and Stefan Mazurkiewicz · See more »
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).
New!!: Banach space and Sublinear function · See more »
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).
New!!: Banach space and Surjective function · See more »
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.
New!!: Banach space and Taylor's theorem · See more »
Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
New!!: Banach space and Tensor product · See more »
Timothy Gowers
Sir William Timothy Gowers, (born 20 November 1963) is a British mathematician.
New!!: Banach space and Timothy Gowers · See more »
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.
New!!: Banach space and Topological vector space · See more »
Total variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure.
New!!: Banach space and Total variation · See more »
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.
New!!: Banach space and Tychonoff's theorem · See more »
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
New!!: Banach space and Uniform boundedness principle · See more »
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure.
New!!: Banach space and Uniform space · See more »
Uniformly convex space
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces.
New!!: Banach space and Uniformly convex space · See more »
Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.
New!!: Banach space and Unit sphere · See more »
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.
New!!: Banach space and Universal property · See more »
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.
New!!: Banach space and Vector space · See more »
Wacław Sierpiński
Wacław Franciszek Sierpiński (14 March 1882 – 21 October 1969) was a Polish mathematician.
New!!: Banach space and Wacław Sierpiński · See more »
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.
New!!: Banach space and Weak topology · See more »
Wiener algebra
In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by, is the space of absolutely convergent Fourier series.
New!!: Banach space and Wiener algebra · See more »
Redirects here:
Banach Space, Banach Spaces, Banach function space, Banach norm, Banach spaces, Complete normed space, Complete normed vector space, Linear Algebra/Banach Spaces.
References
[1] https://en.wikipedia.org/wiki/Banach_space
