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Convex Series
In mathematics, particularly in functional analysis and convex analysis, a is a series of the form \sum_^ r_i x_i where x_1, x_2, \ldots are all elements of a topological vector space X, and all r_1, r_2, \ldots are non-negative real numbers that sum to 1 (that is, such that \sum_^ r_i = 1). Types of Convex series Suppose that S is a subset of X and \sum_^ r_i x_i is a convex series in X. * If all x_1, x_2, \ldots belong to S then the convex series \sum_^ r_i x_i is called a with elements of S. * If the set \left\ is a (von Neumann) bounded set then the series called a . * The convex series \sum_^ r_i x_i is said to be a if the sequence of partial sums \left(\sum_^n r_i x_i\right)_^ converges in X to some element of X, which is called the . * The convex series is called if \sum_^ r_i x_i is a Cauchy series, which by definition means that the sequence of partial sums \left(\sum_^n r_i x_i\right)_^ is a Cauchy sequence. Types of subsets Convex series allow for the de ... [...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] |
Convex Series
In mathematics, particularly in functional analysis and convex analysis, a is a series of the form \sum_^ r_i x_i where x_1, x_2, \ldots are all elements of a topological vector space X, and all r_1, r_2, \ldots are non-negative real numbers that sum to 1 (that is, such that \sum_^ r_i = 1). Types of Convex series Suppose that S is a subset of X and \sum_^ r_i x_i is a convex series in X. * If all x_1, x_2, \ldots belong to S then the convex series \sum_^ r_i x_i is called a with elements of S. * If the set \left\ is a (von Neumann) bounded set then the series called a . * The convex series \sum_^ r_i x_i is said to be a if the sequence of partial sums \left(\sum_^n r_i x_i\right)_^ converges in X to some element of X, which is called the . * The convex series is called if \sum_^ r_i x_i is a Cauchy series, which by definition means that the sequence of partial sums \left(\sum_^n r_i x_i\right)_^ is a Cauchy sequence. Types of subsets Convex series allow for the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence N_1, N_2, \ldots of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with N_i contained in N. Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods. Examples and counterexamples The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers form a countable local base at x. An example of a space that is not first-countable is the cofinite topology on an uncountable set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which Comparison of topologies, agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a Product (category theory), categorical product of its factors, whereas the box topology is too Comparison of topologies, fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of A Multifunction
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve. The graphical representation of the graph of a function is also known as a ''plot''. In the case of functions of two variables – that is, functions whose domain consists of pairs (x, y) –, the graph usually refers to the set of ordered triples (x, y, z) where f(x,y) = z. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface, which can be visualized as a '' surface plot''. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see ''Plot (graphics)'' for details. A graph of a function is a special case of a relation. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivalued Function
In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions, but English Wikipedia currently does, having a separate article for each. A ''multivalued function'' of sets ''f : X → Y'' is a subset : \Gamma_f\ \subseteq \ X\times Y. Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γf''. If ''f'' is an ordinary function, it is a multivalued function by taking its graph : \Gamma_f\ =\ \. They are called single-valued functions to distinguish them. Motivation The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f(z) in some neighbourhood of a point z=a. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B-convex Series
In mathematics, particularly in functional analysis and convex analysis, a is a series of the form \sum_^ r_i x_i where x_1, x_2, \ldots are all elements of a topological vector space X, and all r_1, r_2, \ldots are non-negative real numbers that sum to 1 (that is, such that \sum_^ r_i = 1). Types of Convex series Suppose that S is a subset of X and \sum_^ r_i x_i is a convex series in X. * If all x_1, x_2, \ldots belong to S then the convex series \sum_^ r_i x_i is called a with elements of S. * If the set \left\ is a (von Neumann) bounded set then the series called a . * The convex series \sum_^ r_i x_i is said to be a if the sequence of partial sums \left(\sum_^n r_i x_i\right)_^ converges in X to some element of X, which is called the . * The convex series is called if \sum_^ r_i x_i is a Cauchy series, which by definition means that the sequence of partial sums \left(\sum_^n r_i x_i\right)_^ is a Cauchy sequence. Types of subsets Convex series allow for the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 < r < 1 is real and with then Throughout, will be a map valued in the Extended real number line, extended real numbers with a Domain of a function, domain that is a convex subset of some vector space. The map is a if holds for any real and any |
Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
