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Regulated Integral
In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonné. Definition Definition on step functions Let 'a'', ''b''be a fixed closed, bounded interval in the real line R. A real-valued function ''φ'' : 'a'', ''b''→ R is called a step function if there exists a finite partition :\Pi = \ of 'a'', ''b''such that ''φ'' is constant on each open interval (''t''''i'', ''t''''i''+1) of Π; suppose that this constant value is ''c''''i'' ∈ R. Then, define the integral of a step function ''φ'' to be :\int_a^b \varphi(t) \, \mathrm t := \sum_^ c_i , t_ - t_i , . It can be shown that this definition is independent of the choice of partition, in that if Π1 is another partition of 'a'', ''b''such that ''φ'' is constant on the open intervals of Π1, 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] |
Bounded Linear Operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called " bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the ''area under the curve'' could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Vector 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] |
Mutatis Mutandis
''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used in many countries to acknowledge that a comparison being made requires certain obvious alterations, which are left unstated. It is not to be confused with the similar '' ceteris paribus'', which excludes any changes other than those explicitly mentioned. ''Mutatis mutandis'' is still used in law, economics, mathematics, linguistics and philosophy. In particular, in logic, it is encountered when discussing counterfactuals, as a shorthand for all the initial and derived changes which have been previously discussed. Latin The phrase '—now sometimes written ' to show vowel length—does not appear in surviving classical literature. It is Medieval Latin''Oxford English Dictionary'', 3rd ed. 'mutatis mutandis, ''adv. Oxford University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
