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Bochner Space
In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to functions whose values lie in a Banach space which is not necessarily the space \R or \Complex of real or complex numbers. The space L^p(X) consists of (equivalence classes of) all Bochner measurable functions f with values in the Banach space X whose norm \, f\, _X lies in the standard L^p space. Thus, if X is the set of complex numbers, it is the standard Lebesgue L^p space. Almost all standard results on L^p spaces do hold on Bochner spaces too; in particular, the Bochner spaces L^p(X) are Banach spaces for 1 \leq p \leq \infty. Bochner spaces are named for the mathematician Salomon Bochner. Definition Given a measure space (T, \Sigma; \mu), a Banach space \left(X, \, \,\cdot\,\, _X\right) and 1 \leq p \leq \infty, the Bochner space L^p(T; X) is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions u : T \to X such that 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] |
