# Category: Banach spaces

Clarkson's inequalities
In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions
Infinite-dimensional holomorphy
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values i
Semi-reflexive space
In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (whic
Differentiable vector–valued functions from Euclidean space
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domain
Banach bundle (non-commutative geometry)
In mathematics, a Banach bundle is a fiber bundle over a topological Hausdorff space, such that each fiber has the structure of a Banach space.
Hanner's inequalities
In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces
Multipliers and centralizers (Banach spaces)
In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem.
Strongly measurable function
Strong measurability has a number of different meanings, some of which are explained below.
Uniformly convex space
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.
Goldstine theorem
In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, is stated as follows: Goldstine theorem. Let be a Banach space, then the image of the closed unit
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. A norm is the formalization and the generalization to real vector
Pseudo-monotone operator
In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator. Many problems in t
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the
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
Fredholm kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral e
Schauder basis
In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for
Uniformly smooth space
In mathematics, a uniformly smooth space is a normed vector space satisfying the property that for every there exists such that if with and then The modulus of smoothness of a normed space X is the fu
Ba space
In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is If Σ is a sigm
Besov space
In mathematics, the Besov space (named after Oleg Vladimirovich Besov) is a complete quasinormed space which is a Banach space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel
BK-space
In functional analysis and related areas of mathematics, a BK-space or Banach coordinate space is a sequence space endowed with a suitable norm to turn it into a Banach space. All BK-spaces are normab
Souček space
In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician . One of their main advantages is that they offer a way to deal with the fact that the Sobolev
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the c
Aubin–Lions lemma
In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of
Crinkled arc
In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos. Specifically, consider where is a Hilbert s
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single rea
Modulus and characteristic of convexity
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relat
Grothendieck space
In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space in which every sequence in its continuous dual space that converges in the weak-* topology (also known as th
Browder–Minty theorem
In mathematics, the Browder–Minty theorem (sometimes called the Minty–Browder theorem) states that a bounded, continuous, coercive and monotone function T from a real, separable reflexive Banach space
B-convex space
In functional analysis, the class of B-convex spaces is a class of Banach space. The concept of B-convexity was defined and used to characterize Banach spaces that have the strong law of large numbers
Modulation space
Modulation spaces are a family of Banach spaces defined by the behavior of the short-time Fourier transform withrespect to a test function from the Schwartz space. They were originally proposed by Han
Tsirelson space
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is refle
Orlicz space
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are B
Banach bundle
In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.
Infinite-dimensional Lebesgue measure
In mathematics, there is a theorem stating that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite-dimensional spa
L-infinity
In mathematics, , the (real or complex) vector space of bounded sequences with the supremum norm, and , the vector space of essentially bounded measurable functions with the essential supremum norm, a
In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the sec
Ehrling's lemma
In mathematics, Ehrling's lemma, also known as Lions' lemma, is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev sp
Reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from into its bidual (which is the
Banach manifold
In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more i
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. In essen
Closed range theorem
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.
Opial property
In mathematics, the Opial property is an abstract property of Banach spaces that plays an important role in the study of weak convergence of iterates of mappings of Banach spaces, and of the asymptoti
James' space
In the area of mathematics known as functional analysis, James' space is an important example in the theory of Banach spaces and commonly serves as useful counterexample to general statements concerni
Auerbach's lemma
In mathematics, Auerbach's lemma, named after Herman Auerbach, is a theorem in functional analysis which asserts that a certain property of Euclidean spaces holds for general finite-dimensional normed
Milman–Pettis theorem
In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani ga
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.
Interpolation space
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that ha
Girth (functional analysis)
In functional analysis, the girth of a Banach space is the infimum of lengths of centrally symmetric simple closed curves in the unit sphere of the space. Equivalently, it is twice the infimum of dist
Lorentz space
In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar spaces. The Lorentz spaces are denoted by . Like the spaces, they are c
Fichera's existence principle
In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954. M
Mazur's lemma
In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach space has a sequence of convex combinations of its members that conve
Asplund space
In mathematics — specifically, in functional analysis — an Asplund space or strong differentiability space is a type of well-behaved Banach space. Asplund spaces were introduced in 1968 by the mathema
Infinite-dimensional vector function
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in mos
Riesz–Thorin theorem
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named a
Bs space
In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers or complex numbers such that is finite. The set of such sequences forms a normed
Krein–Smulian theorem
In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark
List of Banach spaces
In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to
Polynomially reflexive space
In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional Mn of degree n (that is, M
Maharam's theorem
In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete me
Dunford–Pettis property
In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another
Method of continuity
In the mathematics of Banach spaces, the method of continuity provides sufficient conditions for deducing the invertibility of one bounded linear operator from that of another, related operator.
Quasi-derivative
In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux der
Uniform norm
In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions defined on a set the non-negative number This norm is also called the supremum norm, the C
C space
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences of real numbers or complex numbers. When equipped with the uniform norm: the sp
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henr
Lp sum
In mathematics, and specifically in functional analysis, the Lp sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its
Quotient of subspace theorem
In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. Let (X, ||·||) be an N-dimensional normed space. There exist
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. The converse is always
Differentiation in Fréchet spaces
In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is
Operator space
In functional analysis, a discipline within mathematics, an operator space is a normed vector space (not necessarily a Banach space) "given together with an isometric embedding into the space B(H) of