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Delta-convergence
In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to (but distinct from) the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim,T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182. and, soon after, under the name of ''almost convergence,'' by Tadeusz Kuczumow. Definition A sequence (x_k) in a metric space (X,d) is said to be Δ-convergent to x\in X if for every y\in X, \limsup(d(x_k,x)-d(x_k,y))\le 0. Characterization in Banach spaces If X is a uniformly convex and uniformly smooth Banach space, with the duality mapping x\mapsto x^* given by \, x\, =\, x^*\, , \langle x^*,x\rangle=\, x\, ^2, then a sequenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opial Property
In mathematics, the Opial property is an abstract property of Banach spaces that plays an important role in the study of weak convergence of iterates of mappings of Banach spaces, and of the asymptotic behaviour of nonlinear semigroups. The property is named after the Polish mathematician Zdzisław Opial. Definitions Let (''X'', , , , , ) be a Banach space. ''X'' is said to have the Opial property if, whenever (''x''''n'')''n''∈N is a sequence in ''X'' converging weakly to some ''x''0 ∈ ''X'' and ''x'' ≠ ''x''0, it follows that :\liminf_ \, x_ - x_ \, < \liminf_ \, x_ - x \, . Alternatively, using the contrapositive, this condition may be written as : If ''X'' is the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Alaoglu Theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact. This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states. History According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a "very important result - maybe most important fact about the weak-* topology - hatechos throughout functional analysis." In 1912, Helly proved that the unit ball of the continuous dual space of C(, b is countably weak-* compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. History Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Convex Space
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a normed vector space such that, for every 00 such that for any two vectors with \, x\, = 1 and \, y\, = 1, the condition :\, x-y\, \geq\varepsilon implies that: :\left\, \frac\right\, \leq 1-\delta. Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. Properties * The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space X is uniformly convex if and only if for every 00 so that, for any two vectors x and y in the closed unit ball (i.e. \, x\, \le 1 and \, y\, \le 1 ) with \, x-y\, \ge \varepsilon , one has \left\, \right\, \le 1-\delta (note that, given \varepsilon , the corresponding value o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Smooth Space
In mathematics, a uniformly smooth space is a normed vector space X satisfying the property that for every \epsilon>0 there exists \delta>0 such that if x,y\in X with \, x\, =1 and \, y\, \leq\delta then :\, x+y\, +\, x-y\, \le 2 + \epsilon\, y\, . The modulus of smoothness of a normed space ''X'' is the function ρ''X'' defined for every by the formula : \rho_X(t) = \sup \Bigl\. The triangle inequality yields that . The normed space ''X'' is uniformly smooth if and only if tends to 0 as ''t'' tends to 0. Properties * Every uniformly smooth Banach space is reflexive. * A Banach space X is uniformly smooth if and only if its continuous dual X^* is uniformly convex (and vice versa, via reflexivity). The moduli of convexity and smoothness are linked by ::\rho_(t) = \sup \, \quad t \ge 0, :and the maximal convex function majorated by the modulus of convexity δ''X'' is given by ::\tilde \delta_X(\varepsilon) = \sup \. :Furthermore, ::\delta_X(\varepsilon / 2) \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Center
In geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball of Q. In the field of parameter estimation, the Chebyshev center approach tries to find an estimator \hat x for x given the feasibility set Q , such that \hat x minimizes the worst possible estimation error for x (e.g. best worst case). Mathematical representation There exist several alternative representations for the Chebyshev center. Consider the set Q and denote its Chebyshev center by \hat. \hat can be computed by solving: : \min_ \left\ with respect to the Euclidean norm \, \cdot\, , or alternatively by solving: : \operatorname \max_ \left\, x - \hat x \right\, ^2. Despite these properties, finding the Chebyshev center may be a hard numerical optimization problem. For example, in the second representation above, the inner maximization i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Functional Analysis
In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Functional Analysis
Nonlinear functional analysis is a branch of mathematical analysis that deals with nonlinear mappings. Topics Its subject matter includes: * generalizations of calculus to Banach spaces * implicit function theorems * fixed-point theorems (Brouwer fixed point theorem, Fixed point theorems in infinite-dimensional spaces, topological degree theory, Jordan separation theorem, Lefschetz fixed-point theorem) * Morse theory and Lusternik–Schnirelmann category theory * methods of complex function theory See also * 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 ... Notes {{Authority control ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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