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B. J. Pettis
Billy James Pettis (1913 – 14 April 1979), was an American mathematician, known for his contributions to functional analysis. See also *Dunford–Pettis property *Dunford–Pettis theorem In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. Measure-theoretic definition Uniform integrability is an extension to the ... * Milman–Pettis theorem * Orlicz–Pettis theorem * Pettis integral * Pettis theorem References *Graves, William H.; Davis, Robert L.; Wright, Fred B., ''Introduction''. In: Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), pp. vii—ix, Contemporary Mathematics 2, Amer. Math. Soc., Providence, R.I., 1980. (This is an introduction to the collection of papers dedicated to the memory of B. J. Pettis.) External links *A Guide to the B. J. Pettis Papers, 1938-19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. 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, for example, continuous function, continuous or unitary operator, unitary 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 v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dunford–Pettis Property
In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous. Many standard Banach spaces have this property, most notably, the space C(K) of continuous functions on a compact space and the space L^1(\mu) of the Lebesgue integrable functions on a measure space. Alexander Grothendieck introduced the concept in the early 1950s , following the work of Dunford and Pettis, who developed earlier results of Shizuo Kakutani, Kōsaku Yosida, and several others. Important results were obtained more recently by Jean Bourgain. Nevertheless, the Dunford–Pettis property is not completely understood. Definition A Banach space X has the Dunford–Pettis property if every continuous weakly compact operator T : X \to Y from X into another Banach space Y transforms weakly compact sets in X into norm-compact sets in Y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dunford–Pettis Theorem
In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in L_1 which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition A: Let (X,\mathfrak, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if \sup_\, f\, _0 there corresponds a \delta>0 such that : \int_E , f, \, d\mu 0 there exists K\in X, I_)\le\varepsilon\ \text X \in \mathcal, where I_ is the indicator function I_ = \begin 1 &\text , X, \geq K, \\ 0 &\text , X, < K. \end. Tightness and uniform integrability Another concept associated with uniform integrability is that of tightness. In this article tightness is taken in a more ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milman–Pettis Theorem
In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 reflexive. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959. Mahlon M. Day (1941) gave examples of reflexive Banach spaces which are not isomorphic to any uniformly convex space. References * S. Kakutani, ''Weak topologies and regularity of Banach spaces'', Proc. Imp. Acad. Tokyo 15 (1939), 169–173. * D. Milman, ''On some criteria for the regularity of spaces of type (B)'', C. R. (Doklady) Acad. Sci. U.R.S.S, 20 (1938), 243–246. * B. J. Pettis, ''A proof that every uniformly convex space is reflexiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pettis Integral
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral. Definition Let f : X \to V where (X,\Sigma,\mu) is a measure space and V is a topological vector space (TVS) with a continuous dual space V' that separates points (that is, if x \in V is nonzero then there is some l \in V' such that l(x) \neq 0), for example, V is a normed space or (more generally) is a Hausdorff locally convex TVS. Evaluation of a functional may be written as a duality pairing: \langle \varphi, x \rangle = \varphi The map f : X \to V is called if for all \varphi \in V', the scalar-valued map \varp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pettis Theorem
In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree. Definition If (X, \Sigma) is a measurable space and B is a Banach space over a field \mathbb (which is the real numbers \R or complex numbers \Complex), then f : X \to B is said to be weakly measurable if, for every continuous linear functional g : B \to \mathbb, the function g \circ f \colon X \to \mathbb \quad \text \quad x \mapsto g(f(x)) is a measurable function with respect to \Sigma and the usual Borel \sigma-algebra on \mathbb. A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space B). Thus, as a special case of the above definition, if (\Omega, \mathca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1979 Deaths
Events January * January 1 ** United Nations Secretary-General Kurt Waldheim heralds the start of the ''International Year of the Child''. Many musicians donate to the ''Music for UNICEF Concert'' fund, among them ABBA, who write the song ''Chiquitita'' to commemorate the event. ** In 1979, the United States officially severed diplomatic ties with the Republic of China (Taiwan). This decision marked a significant shift in U.S. foreign policy, turning to view the People's Republic of China as the sole legitimate representative of China. ** The United States and the People's Republic of China establish full Sino-American relations, diplomatic relations. ** Following a deal agreed during 1978, France, French carmaker Peugeot completes a takeover of American manufacturer Chrysler's Chrysler Europe, European operations, which are based in United Kingdom, Britain's former Rootes Group factories, as well as the former Simca factories in France. * January 6 – Geylang Bahru family ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1913 Births
Events January * January – Joseph Stalin travels to Vienna to research his ''Marxism and the National Question''. This means that, during this month, Stalin, Hitler, Trotsky and Tito are all living in the city. * January 3 – First Balkan War: Greece completes its Battle of Chios (1912), capture of the eastern Aegean island of Chios, as the last Ottoman forces on the island surrender. * 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 18 – 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 23 – 1913 Ottoman coup d'Ă©tat: Enver Pasha comes to power. February * February 1 – New York City's Grand Central Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
