In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
In mathematics and physics, a vector space is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or complex coordinate 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 Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
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 into its bidual is a homeomorphism. A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this evaluation map is an isometric isomorphism and the normed space is a Banach space. Those space for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm of a linear map is the maximum factor by which it "lengthens" vectors.
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all
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. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
In mathematical analysis, a Banach limit is a continuous linear functional :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space of all bounded complex-valued sequences such that for all sequences , in , and complex numbers :
- (linearity);
- if for all , then (positivity);
- , where is the shift operator defined by (shift-invariance);
- if is a convergent sequence, then .
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.
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 Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces.
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector 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
In mathematics, , the vector space of bounded sequences with the supremum norm, and , the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces of absolutely summable sequences, and of absolutely integrable measurable functions. Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras.
In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below, under which it forms a Banach space. Orlicz sequence spaces generalize the spaces, and as such play an important role in functional analysis. Orlicz sequence spaces are particular examples of Orlicz spaces.
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).
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 domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.
This is a glossary for the terminology in a mathematical field of functional analysis.
