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σ-unital Algebra
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net (mathematics), net in a Banach algebra or ring (mathematics), ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert ae_\lambda - a \rVert = 0. Similarly, a left approximate identity in a Banach algebra ''A'' is a net \ such that for every element ''a'' of ''A'', \lim_\lVert e_\lambda a - a \rVert = 0. An approximate identity is a net which is both a right approximate identity and a left approximate identity. C*-algebras For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive elements in ''A'' of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the canonical approxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. As explained below, the spectrum ''Â'' is also naturally a topological space; this is similar to the notion of the spectrum of a ring. One of the most important applications of this concept is to provide a notion of dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nascent Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are particular smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a (generalized) function, convolving it with a mollifier "mollifies" it, that is, its sharp features are smoothed, while still remaining close to the original. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them. Historical notes Mollifiers were introduced by Kurt Otto Friedrichs in his paper , which is considered a watershed in the modern theory of partial differential equations.See the commentary of Peter Lax on the paper in . The name of this mathematical object has a curious genesis, and Peter Lax tells the story in his commentary on that paper published in Friedrichs' "''Selecta''". According to him, at that time, the mathematician Donald Alexander Flan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fejér Kernel
In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesà ro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungary, Hungarian mathematician Lipót Fejér (1880–1959). Definition The Fejér kernel has many equivalent definitions. We outline three such definitions below: 1) The traditional definition expresses the Fejér kernel F_n(x) in terms of the Dirichlet kernel: F_n(x) = \frac \sum_^D_k(x) where :D_k(x)=\sum_^k ^ is the ''k''th order Dirichlet kernel. 2) The Fejér kernel F_n(x) may also be written in a closed form expression as follows F_n(x) = \frac \left(\frac\right)^2 = \frac \left(\frac\right) This closed form expression may be derived from the definitions used above. The proof of this result goes as follows. First, we use the fact that the Dirichlet kernel may be written as: :D_k(x)=\frac Hence, using the definition of the Fejér kernel above we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
