|
Riesz–Fischer Theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces L^p from Lebesgue integration theory are complete. Modern forms of the theorem The most common form of the theorem states that a measurable function on \pi, \pi/math> is square integrable if and only if the corresponding Fourier series converges in the Lp space L^2. This means that if the ''N''th partial sum of the Fourier series corresponding to a square-integrable function ''f'' is given by S_N f(x) = \sum_^ F_n \, \mathrm^, where F_n, the ''n''th Fourier coefficient, is given by F_n =\frac\int_^\pi f(x)\, \mathrm^\, \mathrmx, then \lim_ \left\Vert S_N f - f \right\, _2 = 0, where \, \,\cdot\,\, _2 is the L^2-norm. Conve ... [...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] |
Bessel's Inequality
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let H be a Hilbert space, and suppose that e_1, e_2, ... is an orthonormal sequence in H. Then, for any x in H one has :\sum_^\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2, where ⟨·,·⟩ denotes the inner product in the Hilbert space H. If we define the infinite sum :x' = \sum_^\left\langle x,e_k\right\rangle e_k, consisting of "infinite sum" of vector resolute x in direction e_k, Bessel's inequality tells us that this series converges. One can think of it that there exists x' \in H that can be described in terms of potential basis e_1, e_2, \dots. For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Inequality
In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the triangle inequality :\, f+g\, _p \le \, f\, _p + \, g\, _p with equality for if and only if ''f'' and ''g'' are positively linearly dependent, i.e., for some or . Here, the norm is given by: :\, f\, _p = \left( \int , f, ^p d\mu \right)^ if ''p'' q, then \, f\, _\leq\, f\, _. Reverse inequality When p< 1 the reverse inequality holds: : We further need the restriction that both and are non-negative, as we can see from the example and : . The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Sequence
In 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 ..., a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose Element (mathematics), elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: a_n=\sqrt n, the consecutive terms become arbitrarily close to each other: a_-a_n = \sqrt-\sqrt = \frac d. (Actually, any m > \left(\sqrt + d\right)^2 suffices.) As a result, despite how far one goes, the remaining terms of the sequence never get c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It can also re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
