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Eberlein–Šmulian Theorem
In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in a Banach space. Statement Eberlein–Šmulian theorem: If ''X'' is a Banach space and ''A'' is a subset of ''X'', then the following statements are equivalent: # each sequence of elements of ''A'' has a subsequence that is weakly convergent in ''X'' # each sequence of elements of ''A'' has a weak cluster point in ''X'' # the weak closure of ''A'' is weakly compact. A set ''A'' can be weakly compact in three different ways: * Sequential compactness: Every sequence from ''A'' has a convergent subsequence whose limit is in ''A''. * Limit point compactness: Every infinite subset of ''A'' has a limit point in ''A''. * Compactness (or Heine- Borel compactness): Every open cover of ''A'' admits a finite subcover. The Eberlein–Šmulian theorem states that ... [...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 t ... [...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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Spaces
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 complete n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldstine Theorem
In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, is stated as follows: :Goldstine theorem. Let X be a Banach space, then the image of the closed unit ball B \subseteq X under the canonical embedding into the closed unit ball B^ of the bidual space X^ is a weak*-dense subset. The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space of real sequences that converge to zero, c0 space c_0, and its bi-dual space Lp space \ell^. Proof Lemma For all x^ \in B^, \varphi_1, \ldots, \varphi_n \in X^ and \delta > 0, there exists an x \in (1+\delta)B such that \varphi_i(x) = x^(\varphi_i) for all 1 \leq i \leq n. Proof of lemma By the surjectivity of \begin \Phi : X \to \Complex^, \\ x \mapsto \left(\varphi_1(x), \cdots, \varphi_n(x) \right) \end it is possible to find x \in X with \varphi_i(x) = x^(\varphi_i) for 1 \leq i \leq n. Now let Y := \bigcap_i \ker \varphi_i = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James' Theorem
In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X is reflexive if and only if every continuous linear functional on X attains its supremum on the closed unit ball in X. A stronger version of the theorem states that a weakly closed subset C of a Banach space X is weakly compact if and only if each continuous linear functional on X attains a maximum on C. The hypothesis of completeness in the theorem cannot be dropped. Statements The space X considered can be a real or complex Banach space. Its continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ... is denoted by X^. The topological dual of ℝ-Banach space deduced from X by any restriction scalar will be denoted X^_. (It is of interest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mazur's Lemma
In mathematics, Mazur's lemma is a result in the theory of normed vector spaces. It shows that any weakly convergent sequence in a normed space has a sequence of convex combinations of its members that converges strongly to the same limit, and is used in the proof of Tonelli's theorem. Statement of the lemma Let (X, \, \,\cdot\,\, ) be a normed vector space and let \left(u_n\right)_ be a sequence in X that converges weakly to some u_0 in X: u_n \rightharpoonup u_0 \mbox n \to \infty. That is, for every continuous linear functional f \in X^, the continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ... of X, f\left(u_n\right) \to f\left(u_0\right) \mbox n \to \infty. Then there exists a function N : \N \to \N and a sequence of sets of real numbers \left\ such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bishop–Phelps Theorem
In mathematics, the Bishop–Phelps theorem is a theorem about the topological properties of Banach spaces named after Errett Bishop and Robert Phelps, who published its proof in 1961. Statement Importantly, this theorem fails for complex Banach spaces. However, for the special case where B is the closed unit ball then this theorem does hold for complex Banach spaces. See also * * * * * * References {{DEFAULTSORT:Bishop-Phelps theorem Banach spaces Theorems in functional analysis ... [...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-* compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
