B. J. Pettis

	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 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, 1938-1980 1979 deaths 20th-century American mathematicians 1913 births {{US-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 were Thales of Miletus (c. 624–c.546 BC); 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' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagoreans, 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 mathemat ... [...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, norm, topology, etc.) and the 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 spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the ... [...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(''μ'') 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'' → ''Y'' from ''X'' into another Banach space ''Y'' transforms weakly co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 such that : \sup_\int_A, f, \, d\mu 0 such that, for every measurable A such that P(A)\leq \delta and every X in \mathcal, \operatorname E(, X, I_A)\leq\varepsilon. or alternatively 2. A class \mathcal of random variables is called uniformly integrable (UI) if there exists K\in X, I_)\le\varepsilon\ \text X \in \mathcal, where I_ is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Milman–Pettis Theorem In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman (1938) and 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 Milman–Pettis theorem Orlicz–Pettis theorem Petti ... (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]
	Orlicz–Pettis Theorem A theorem in functional analysis concerning convergent series (Orlicz) or, equivalently, countable additivity of measures (Pettis) with values in abstract spaces. Let X be a Hausdorff locally convex topological vector space with dual X^. A series \sum_^\infty~x_n is ''subseries convergent'' (in X ), if all its subseries \sum_^\infty~ x_ are convergent. The theorem says that, equivalently, (i) If a series \sum_^\infty~x_n is weakly subseries convergent in X (i.e., is subseries convergent in X with respect to its weak topology \sigma(X,X^)), then it is (subseries) convergent; or (ii) Let \mathbf be a \sigma-algebra of sets and let \mu:\mathbf\to X be an additive set function. If \mu is weakly countably additive, then it is countably additive (in the original topology of the space X ). The history of the origins of the theorem is somewhat complicated. In numerous papers and books there are misquotations or/and misconceptions concerning the result. Assuming t ... [...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 Vis 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 \varphi ... [...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, \mathc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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. 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 7 – Cambodian–Vietnamese War: The People's Army of Vietnam and Vietnamese-backed Kampuchean United Front for National Salvation, Cambodian insurgents announce the fall of Phnom Penh, Cambodia, and the collapse of the Pol Pot regime. Pol Pot and the Khmer Rouge retreat west to an area ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]