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Freudenthal Spectral Theorem
In mathematics, the Freudenthal spectral theorem is a result in Riesz space theory proved by Hans Freudenthal in 1936. It roughly states that any element dominated by a positive element in a Riesz space with the principal projection property can in a sense be approximated uniformly by simple functions. Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-known Radon–Nikodym theorem, the validity of the Poisson formula and the spectral theorem from the theory of normal operators can all be shown to follow as special cases of the Freudenthal spectral theorem. Statement Let ''e'' be any positive element in a Riesz space ''E''. A positive element of ''p'' in ''E'' is called a component of ''e'' if p\wedge(e-p)=0. If p_1,p_2,\ldots,p_n are pairwise disjoint components of ''e'', any real linear combination of p_1,p_2,\ldots,p_n is called an ''e''-simple function. The Freudenthal spectral theorem states: Let ''E'' be any Riesz space with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space. A measurable space consists of the first two components without a specific measure. Definition A measure space is a triple (X, \mathcal A, \mu), where * X is a set * \mathcal A is a -algebra on the set X * \mu is a measure on (X, \mathcal) In other words, a measure space consists of a measurable space (X, \mathcal) together with a measure on it. Example Set X = \. The \sigma-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by \wp(\cdot). Sticking with this convention, we set \mathcal = \wp(X) In this simple case, the power set can be written down ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Convergence Theorem
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum. Convergence of a monotone sequence of real numbers Lemma 1 If a sequence of real numbers is increasing and bounded above, then its supremum is the limit. Proof Let (a_n)_ be such a sequence, and let \ be the set of terms of (a_n)_ . By assumption, \ is non-empty and bounded above. By the least-upper-bound property of real numbers, c = \sup_n \ exists and is finite. Now, for every \varepsilon > 0, there exists N such that a_N > c - \varepsilon , since otherwise c - \varepsilon is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Norm
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'', ''b''⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation ''x'' ↦ ''f''(''x''), for ''x'' ∈ 'a'', ''b'' Functions whose total variation is finite are called functions of bounded variation. Historical note The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper . He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons. Definitions Total variation for functions of one real variable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Lattice
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, such that for all , the implication \Rightarrow holds, where the absolute value is defined as , x, = x \vee -x\text Examples and constructions Banach lattices are extremely common in functional analysis, and "every known example n 1948of a Banach space asalso a vector lattice." In particular: * , together with its absolute value as a norm, is a Banach lattice. * Let be a topological space, a Banach lattice and the space of continuous bounded functions from to with norm \, f\, _ = \sup_ \, f(x)\, _Y\text Then is a Banach lattice under the pointwise partial order: \Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text Examples of non-lattice Banach spaces are now known; James' space is one such.Kania, Tomasz (12 April 2017).Answerto "Banach space that is not a Banach lattice" (accessed 13 August 2022). ''Mathematics St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Complete
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if every non-empty subset of with an upper bound has a ''least'' upper bound (supremum) in . Not every (partially) ordered set has the least upper bound property. For example, the set \mathbb of all rational numbers with its natural order does ''not'' have the least upper bound property. The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness.Willard says that an ordered space "X is Dedekind complete if every subset of X having an upper bound has a least upper bound." (pp. 124-5, Problem 17E.) It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures". Given a measurable space (X, \Sigma) (that is, a set X with a σ-algebra \Sigma on it), an extended signed measure is a set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ... \mu : \Sigma \to \R \cup \ such that \mu(\varnothing) = 0 and \mu is sigma additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur la décomposition des opérations fonctionelles linéaires''. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis. Definition Preliminaries If X is an ordered vector space (which by definition is a vector space over the reals) and if S is a subset of X then an element b \in X is an upper bound (resp. lower bound) of S if s \leq b (resp. s \geq b) for all s \in S. An element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Sur la décomposition des opérations fonctionelles linéaires''. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis. Definition Preliminaries If X is an ordered vector space (which by definition is a vector space over the reals) and if S is a subset of X then an element b \in X is an upper bound (resp. lower bound) of S if s \leq b (resp. s \geq b) for all s \in S. An element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are * unitary operators: ''N*'' = ''N−1'' * Hermitian operators (i.e., self-adjoint operators): ''N*'' = ''N'' * Skew-Hermitian operators: ''N*'' = −''N'' * positive operators: ''N'' = ''MM*'' for some ''M'' (so ''N'' is self-adjoint). A normal matrix is the matrix expression of a normal operator on the Hilbert space C''n''. Properties Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable. Let T be a bounded operator. The following are equivalent. * T is normal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
