|
Distortion Problem
In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space ''X'' is called λ-distortable if there exists an equivalent norm , ''x'', on ''X'' such that, for all infinite-dimensional subspaces ''Y'' in ''X'', :\sup_ \frac \ge \lambda (see distortion (mathematics)). Note that every Banach space is trivially 1-distortable. A Banach space is called distortable if it is λ-distortable for some λ > 1 and it is called arbitrarily distortable if it is λ-distortable for any λ. Distortability first emerged as an important property of Banach spaces in the 1960s, where it was studied by and . James proved that ''c''0 and ℓ1 are not distortable. Milman showed that if ''X'' is a Banach space that does not contain an isomorphic copy of ''c''0 or ℓ''p'' for some (see sequence space), then some infinite-dimensional subspace o ... [...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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. 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 continuous function, continuous, unitary operator, unitary etc. 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 variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distortion (mathematics)
In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal. The distortion of a function ƒ of the plane is given by :H(z,f) = \limsup_\frac which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at ''z''. This geometrical definition is often very difficult to work with, and the necessary analytical features can be extrapolated to the following definition. A mapping ''ƒ'' : Ω → R2 from an open domain in the plane to the plane has finite distortion at a point ''x'' ∈ Ω if ''ƒ'' is in the Sobolev space W(Ω, R2), the Jacobian determinant J(''x'',ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function ''K''('' ... [...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 complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence Space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field ''K'' of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in ''K'', and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the ''p''-norm. These are special cases of L''p'' spaces for the counting measure on the set of natural numbers. Other important classes of sequences like ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Metric Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An importa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Function
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclus ... [...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 that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsirelson Space
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an lp space, ℓ ''p'' space nor a Sequence space#c and c0, ''c''0 space can be embedded. The Tsirelson space is reflexive space, reflexive. It was introduced by B. S. Tsirelson in 1974. The same year, Figiel and Johnson published a related article () where they used the notation ''T'' for the ''dual'' of Tsirelson's example. Today, the letter ''T'' is the standard notationsee for example , p. 8; , p. 95; ''The Handbook of the Geometry of Banach Spaces'', vol. 1, p. 276; vol. 2, p. 1060, 1649. for the dual of the original example, while the original Tsirelson example is denoted by ''T''*. In ''T''* or in ''T'', no subspace is isomorphic, as Banach space, to an ''ℓ'' ''p'' space, 1 ≤ ''p'' < ∞, or to ''c''0. All classical Banach spaces known to , spaces of continuous functions, of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russian Mathematical Surveys
''Uspekhi Matematicheskikh Nauk'' (russian: Успехи математических наук) is a Russian mathematical journal, published by the Russian Academy of Sciences and Moscow Mathematical Society and translated into English as ''Russian Mathematical Surveys''. ''Uspekhi Matematicheskikh Nauk'' was founded in 1936, with Lazar Lyusternik as its editor-in-chief. Initially, it appeared irregularly, with issues devoted to specific topics within mathematics together with non-research articles about the work of different mathematical institutes in Russia and abroad. Its third issue, in 1937, was devoted to attacks on Nikolai Luzin, but in an anniversary issue 24 years later this politicization of the journal was downplayed. After a hiatus for World War II, the journal began publishing on a regular schedule in 1946. Its translation, ''Russian Mathematical Surveys'', began in 1960 and since 1997 has been published jointly by the London Mathematical Society, Turpion Ltd, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric And Functional Analysis (journal)
''Geometric and Functional Analysis'' (''GAFA'') is a mathematical journal published by Birkhäuser, an independent division of Springer-Verlag. The journal is published approximately bi-monthly. The journal publishes papers on broad range of topics in geometry and analysis including geometric analysis, riemannian geometry, symplectic geometry, geometric group theory, non-commutative geometry, automorphic forms and analytic number theory, and others. ''GAFA'' is both an acronym and a part of the official full name of the journal. History ''GAFA'' was founded in 1991 by Mikhail Gromov and Vitali Milman. The idea for the journal was inspired by the long-running Israeli seminar series "Geometric Aspects of Functional Analysis" of which Vitali Milman had been one of the main organizers in the previous years. The journal retained the same acronym as the series to stress the connection between the two. Journal information The journal is reviewed cover-to-cover in Mathematical Review ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals".. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics". Publication history The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees. Poincaré episode The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
