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Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics. His major work was the 1932 book, ''Théorie des opérations linéaires'' (Theory of Linear Operations), the first monograph on the general theory of functional analysis. Born in Kraków to a family of Gorals, Goral descent, Banach showed a keen interest in mathematics and engaged in solving mathematical problems during school Recess (break), recess. After completing his secondary education, he befriended Hugo Steinhaus, with whom he established the Polish Mathematical Society in 1919 and later published the scientific journal ''Studia Mathematica''. In 1920, he received an assistantship at the Lwów Polytechnic, subsequently becoming a professor in 1922 and a member of the Polish Academy of Lear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kraków
, officially the Royal Capital City of Kraków, is the List of cities and towns in Poland, second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city has a population of 804,237 (2023), with approximately 8 million additional people living within a radius. Kraków was the official capital of Poland until 1596, and has traditionally been one of the leading centres of Polish academic, cultural, and artistic life. Cited as one of Europe's most beautiful cities, its Kraków Old Town, Old Town was declared a UNESCO World Heritage Site in 1978, one of the world's first sites granted the status. The city began as a Hamlet (place), hamlet on Wawel Hill and was a busy trading centre of Central Europe in 985. In 1038, it became the seat of King of Poland, Polish monarchs from the Piast dynasty, and subsequently served as the centre of administration under Jagiellonian dynasty, Jagiellonian kings and of the Polish–Lithuan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Tarski Paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their original shape. However, the pieces themselves are not "solids" in the traditional sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox". The theorem is a veridical paradox: ... [...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] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Academy Of Learning
The Polish Academy of Arts and Sciences or Polish Academy of Learning (, PAU), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of sciences (the other being the Polish Academy of Sciences, headquartered in Warsaw). The Polish Academy of Arts and Sciences is co-owner of the Polish Library in Paris. History The Academy traces its origins to Academy of Learning founded in 1871, itself a result of the transformation of the , in existence since 1815. Though formally limited to the Austrian Partition, the Academy served from the beginning as a learned and cultural society for the entire Polish nation. Its activities extended beyond the boundaries of the Austrian Partition, gathering scholars from all of Poland, and many other countries as well. Some indication of how the Academy's influence extended beyond the boundaries of the Partitions came in 1893, when the collection of the Polish Library in Pari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjection Of Fréchet Spaces
The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective. The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal. Preliminaries, definitions, and notation Let L : X \to Y be a continuous linear map between topological vector spaces. The continuous dual space of X is denoted by X^. The transpose of L is the map ^t L : Y^ \to X^ defined by L \left(y^\right) := y^ \circ L. If L : X \to Y is surjective then ^t L : Y^ \to X^ will be injective, but the converse is not true in general. The weak t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Bundle
In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension. Definition of a Banach bundle Let ''M'' be a Banach manifold of class ''C''''p'' with ''p'' ≥ 0, called the base space; let ''E'' be a topological space, called the total space; let ''π'' : ''E'' → ''M'' be a surjective continuous map. Suppose that for each point ''x'' ∈ ''M'', the fibre ''E''''x'' = ''π''−1(''x'') has been given the structure of a Banach space. Let :\ be an open cover of ''M''. Suppose also that for each ''i'' ∈ ''I'', there is a Banach space ''X''''i'' and a map ''τ''''i'' :\tau_ : \pi^ (U_) \to U_ \times X_ such that * the map ''τ''''i'' is a homeomorphism commuting with the projection onto ''U''''i'', i.e. the following diagram commutes: :: : and for each ''x'' ∈ ''U''''i'' the induced map ''τ''''ix'' on the fibre ''E''''x'' ::\tau_ : \pi^ (x) \to X_ : is an invertibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Manifold
In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to infinite dimensions. A further generalisation is to Fréchet manifolds, replacing Banach spaces by Fréchet spaces. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modeled on Hilbert spaces. Definition Let X be a set. An atlas of class C^r, r \geq 0, on X is a collection of pairs (called charts) \left(U_i, \varphi_i\right), i \in I, such that # each U_i is a subset of X and the union of the U_i is the whole of X; # each \varphi_i is a bijection from U_i onto an open subset \varphi_i\left(U_i\right) of some Banach space E_i, and for any indices i \text j, \varphi_i\left(U_i \cap U_j\right) is open in E_i; # t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Stone Theorem
In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space ''X'' from the Banach space structure of the space ''C''(''X'') of continuous real- or complex-valued functions on ''X''. If one is allowed to invoke the algebra structure of ''C''(''X'') this is easy – we can identify ''X'' with the spectrum of ''C''(''X''), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space ''C''(''X'')*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering ''X'' from the extreme points of the unit ball of ''C''(''X'')*. Statement For a compact Hausdorff space ''X'', let ''C''(''X'') denote the Banach space of continuous real- or complex-valued functions on ''X'', equipped with the supremum ... [...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-* co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn–Banach Theorem
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. History The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C , b/math> of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Schauder Theorem
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T^. Statement and proof The proof here uses the Baire category theorem, and completeness of both E and F is essential to the theorem. The statement of the theorem is no longer true if either space is assumed to be only a normed vector space; see . The proof is based on the following lemmas, which are also somewhat of independent interest. A linear map f : E \to F between topological vector spaces is said to be nearly open if, for each n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
