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Hahn–Banach Theorem
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. History The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C[a, b] of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theorem was proved by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Separation Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension Of A Function
In mathematics, the restriction of a function f is a new function, denoted f\vert_A or f , obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert_A. Formal definition Let f : E \to F be a function from a set E to a set F. If a set A is a subset of E, then the restriction of f to A is the function _A : A \to F given by _A(x) = f(x) for x \in A. Informally, the restriction of f to A is the same function as f, but is only defined on A. If the function f is thought of as a relation (x,f(x)) on the Cartesian product E \times F, then the restriction of f to A can be represented by its graph where the pairs (x,f(x)) represent ordered pairs in the graph G. Extensions A function F is said to be an ' of another function f if whenever x is in the domain of f then x is also in the domain of F and f(x) = F(x). That is, if \operatorname f \subseteq \operatorname F and F\big\vert_ = f. A '' '' (respectively, '' '', etc.) of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists of a surjective function from onto , and the indexed collection is typically called an '' (indexed) family'', often written as . Examples *An enumeration of a set gives an index set J \sub \N, where is the particular enumeration of . *Any countably infinite set can be (injectively) indexed by the set of natural numbers \N. *For r \in \R, the indicator function on is the function \mathbf_r\colon \R \to \ given by \mathbf_r (x) := \begin 0, & \mbox x \ne r \\ 1, & \mbox x = r. \end The set of all such indicator functions, \_ , is an uncountable set indexed by \mathbb. Other uses In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm that can sample the set efficiently; e. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1932
Events January * January 4 – The British authorities in India arrest and intern Mahatma Gandhi and Vallabhbhai Patel. * January 9 – Sakuradamon Incident (1932), Sakuradamon Incident: Korean nationalist Lee Bong-chang fails in his effort to assassinate Emperor Hirohito of Japan. The Kuomintang's official newspaper runs an editorial expressing regret that the attempt failed, which is used by the Japanese as a pretext to attack Shanghai later in the month. * January 22 – The 1932 Salvadoran peasant uprising begins; it is suppressed by the government of Maximiliano Hernández Martínez. * January 24 – Marshal Pietro Badoglio declares the end of Libyan resistance. * January 26 – British submarine sinks with all 60 hands. * January 28 – January 28 incident: Conflict between Japan and China in Shanghai. * January 31 – Japanese warships arrive in Nanking. February * February 2 ** A general World Disarmament Conference begins in Geneva. The principal issue at the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1913
Events January * January 5 – First Balkan War: Battle of Lemnos (1913), Battle of Lemnos – Greek admiral Pavlos Kountouriotis forces the Turkish fleet to retreat to its base within the Dardanelles, from which it will not venture for the rest of the war. * January 13 – Edward Carson founds the (first) Ulster Volunteers, Ulster Volunteer Force, by unifying several existing Ulster loyalism, loyalist militias to resist home rule for Ireland. * January 23 – 1913 Ottoman coup d'état: Ismail Enver comes to power. * January – Stalin (whose first article using this name is published this month) travels to Vienna to carry out research. Until he leaves on February 16 the city is home simultaneously to him, Hitler, Trotsky and Josip Broz Tito, Tito alongside Alban Berg, Berg, Freud and Jung and Ludwig Wittgenstein, Ludwig and Paul Wittgenstein. February * February 1 – New York City's Grand Central Terminal, having been rebuilt, reopens as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1910
Events January * January 13 – The first public radio broadcast takes place; live performances of the operas '' Cavalleria rusticana'' and ''Pagliacci'' are sent out over the airwaves, from the Metropolitan Opera House in New York City. February * February 20 – Boutros Ghali, the first native-born Prime Minister of Egypt, is assassinated in Cairo. March * March – Albanian revolt of 1910: An uprising against Ottoman rule breaks out in Albania. * March 8 – In France, Raymonde de Laroche is awarded Pilot's license No. 36 by the Federation Aeronautique Internationale, becoming the first woman authorized to fly an airplane. * March 10 ** Slavery in China, which has existed since the Shang dynasty, is now made illegal. ** Nazareth Baptist Church, an African-initiated church, is founded by Prophet Isaiah Shembe in South Africa. * March 17 – Progressive Republicans in the United States House of Representatives rebel against Speaker Joseph Gurney Cann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Sine And Cosine Series
In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. Notation In this article, denotes a real valued function on \mathbb which is periodic with period 2''L''. Sine series If is an odd function with period 2L, then the Fourier Half Range sine series of ''f'' is defined to be f(x) = \sum_^\infty b_n \sin \frac which is just a form of complete Fourier series with the only difference that a_0 and a_n is zero, and the series is defined for half of the interval. In the formula we have b_n = \frac \int_0^L f(x) \sin \frac \, dx, \quad n \in \mathbb . Cosine series If is an even function with a period 2L, then the Fourier cosine series is defined to be f(x) = \frac + \sum_^ c_n \cos \frac where c_n = \frac \int_0^L f(x) \cos \frac \, dx, \quad n \in \mathbb_0 . Remarks This notion can be generalized to functions which are not even or odd, but then the above formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
