## functional analysis

In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space. ...more on Wikipedia about "Absorbing set"

In functional analysis, a right approximate identity in a Banach algebra A is a net (or a sequence) ...more on Wikipedia about "Approximate identity"

In mathematics, the Arzelà-Ascoli theorem of functional analysis is a criterion to decide whether a set of continuous functions from a compact metric space into a metric space is compact in the topology of uniform convergence. ...more on Wikipedia about "Arzelà-Ascoli theorem"

The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space. ...more on Wikipedia about "Baire category theorem"

In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honour of René-Louis Baire who introduced the concept. ...more on Wikipedia about "Baire space"

In mathematical analysis, a Banach limit is a continuous linear functional $\phi: \ell_\infty \to \mathbb\left\{R\right\}$ defined on the Banach space $\ell_\infty$ of all bounded real-valued sequences such that for any real-valued sequences $x=\left(x_n\right)$ and $y=\left(y_n\right)$, the following conditions are satisfied: ...more on Wikipedia about "Banach limit"

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. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact. ...more on Wikipedia about "Banach-Alaoglu theorem"

In functional analysis and related areas of mathematics a barrelled set or a barrel in a topological vector space is a set which is convex, balanced, absorbing and closed. ...more on Wikipedia about "Barrelled set"

In functional analysis and related areas of mathematics barrelled spaces are topological vector spaces where every barrelled set in the space is a neighbourhood for the zero vector. They are studied because the Banach-Steinhaus theorem still holds for them. ...more on Wikipedia about "Barrelled space"

In functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are functions not vectors. This means that each function in the function space can be represented as a linear combination of the basis functions. ...more on Wikipedia about "Basis function"

In the mathematical theory of functional analysis, Bernstein's inequality is defined as follows. ...more on Wikipedia about "Bernstein's inequality"

In mathematics, Bessel's inequality is a statement about the coefficients of an element $x$ in a Hilbert space in respect to an orthonormal sequence. ...more on Wikipedia about "Bessel's inequality"

In functional analysis and related areas of mathematics, the beta-dual or $\beta$-dual is a certain linear subspace of the algebraic dual of a sequence space. ...more on Wikipedia about "Beta-dual space"

In mathematics, a biorthogonal system in a pair of topological vector spaces E and F that are in duality is a pair of indexed subsets ...more on Wikipedia about "Biorthogonal system"

In functional analysis, the Borel functional calculus is a functional calculus (i.e. an assignment of operators to functions defined on the real line), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function ss2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator -Δ or the exponential ...more on Wikipedia about "Borel functional calculus"

In functional analysis and related areas of mathematics a bornological space or Mackey space, named after George Mackey, is a locally convex vector space where the continuous linear operators to any locally convex vector space are exactly the bounded linear operators. ...more on Wikipedia about "Bornological space"

In functional analysis (a branch of mathematics), 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 by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X, ...more on Wikipedia about "Bounded operator"

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded. ...more on Wikipedia about "Bounded set"

Bred vectors are perturbations, related to Lyapunov vectors, that capture fast growing dynamical instabilities of the solution of a numerical model. They are used, for example, as initial perturbations for ensemble forecasting in numerical weather prediction. ...more on Wikipedia about "Bred vectors"

(C0-semigroup) In mathematics, a C0-semigroup is a continuous morphism from (R+,+) into a topological monoid, usually L(H), the algebra of linear continuous operators on some Hilbert space H. ...more on Wikipedia about "C0-semigroup"

for any f, g in V is called the canonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra when V is finite dimensional is discussed in the Stone-von Neumann theorem. ...more on Wikipedia about "CCR and CAR algebras" It's time to think about www.shortopedia.com.

In mathematics, Choquet theory is an area of functional analysis and convex analysis concerned with measures with support on the extreme points of a convex set C. Roughly speaking, all vectors of C should appear as 'averages' of extreme points, a concept made more precise by the idea of convex combinations replaced by integrals taken over the set E of extreme polints. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional topological vector space along lines similar to the finite-dimensional case. The name is for Gustave Choquet, whose main concerns were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics. ...more on Wikipedia about "Choquet theory"

In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph. ...more on Wikipedia about "Closed graph theorem"

In mathematics, the Colombeau algebra is an algebra introduced with the aim of constructing an improved theory of distributions, in which multiplication is not problematic. The origins of the theory are in applications to quasilinear hyperbolic partial differential equations. ...more on Wikipedia about "Colombeau algebra"

In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator ...more on Wikipedia about "Compression (functional analysis)"