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 mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M.
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.
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.
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 Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu.
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.
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 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, 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.
In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra over the real or complex numbers that at the same time is also a Fréchet space. The multiplication operation for is required to be jointly continuous. If is an increasing family of seminorms for the topology of , the joint continuity of multiplication is equivalent to there being a constant and integer for each such that for all . Fréchet algebras are also called B0-algebras.
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for n ≥ N to ensure the 'difference' between μn and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of R-modules rather than metric spaces.
The Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner. This mathematics concept is due to Pavel Urysohn.
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. The theorem was proved by Mikhail Kadec (1966) and Richard Davis Anderson.
This is a glossary for the terminology in a mathematical field of functional analysis.
In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
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. Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the canonical LF-topology, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called the space of distributions on and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology.
