Functional Analysis
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 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] picture info Drum Vibration Mode12 The drum is a member of the percussion group of musical instruments. In the Hornbostel-Sachs classification system, it is a membranophone. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a shell and struck, either directly with the player's hands, or with a percussion mallet, to produce sound. There is usually a resonant head on the underside of the drum. Other techniques have been used to cause drums to make sound, such as the thumb roll. Drums are the world's oldest and most ubiquitous musical instruments, and the basic design has remained virtually unchanged for thousands of years. Drums may be played individually, with the player using a single drum, and some drums such as the djembe are almost always played in this way. Others are normally played in a set of two or more, all played by the one player, such as bongo drums and timpani. A number of different drums together with cymbals form the basic modern drum kit. Uses ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in Ancona, then part of the Papal States, into a very poor Jewish family: his father was Abramo Volterra and his mother, Angelica Almagià. Abramo Volterra died in 1862 when Vito was two years old. The family moved to Turin, and then to Florence, where he studied at the Dante Alighieri Technical School and the Galileo Galilei Technical Institute. Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics in 1883. He immediately started work developing his theory of functionals which led to his interest and later contributions in integral and integro-differential equations. His work is summarised in his book ''Theory ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Integral In 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 ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Measure (mathematics) In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Linear Algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Springer Publishing Springer Publishing Company is an American publishing company of academic journals and books, focusing on the fields of nursing, gerontology, psychology, social work, counseling, public health, and rehabilitation (neuropsychology). It was established in 1951 by Bernhard Springer, a great-grandson of Julius Springer, and is based in Midtown Manhattan, New York City. History Springer Publishing Company was founded in 1950 by Bernhard Springer, the Berlin-born great-grandson of Julius Springer, who founded Springer-Verlag (now Springer Science+Business Media). Springer Publishing's first landmark publications included ''Livestock Health Encyclopedia'' by R. Seiden and the 1952 ''Handbook of Cardiology for Nurses''. The company's books soon branched into other fields, including medicine and psychology. Nursing publications grew rapidly in number, as Modell's ''Drugs in Current Use'', a small annual paperback, sold over 150,000 copies over several editions. Solomon Garb's ''Labor ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Springer Science & Business Media Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology". Springer Science+Business Media. In 1964, Springer expanded its business internationa ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Dimension (vector Space) In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as : F read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any field F. The complex numbers \Complex are both a real and complex vector space; we have ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info 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 Goral descent, Banach showed a keen interest in mathematics and engaged in solving mathematical problems during school 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 Learning in 1924. Banach ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populous member state of the European Union. Warsaw is the nation's capital and largest metropolis. Other major cities include Kraków, Wrocław, Łódź, Poznań, Gdańsk, and Szczecin. Poland has a temperate transitional climate and its territory traverses the Central European Plain, extending from Baltic Sea in the north to Sudeten and Carpathian Mountains in the south. The longest Polish river is the Vistula, and Poland's highest point is Mount Rysy, situated in the Tatra mountain range of the Carpathians. The country is bordered by Lithuania and Russia to the northeast, Belarus and Ukraine to the east, Slovakia and the Czech Republic to the south, and Germany to the west. It also shares maritime boundaries with Denmark and Sweden. ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Lwów School Of Mathematics The Lwów school of mathematics ( pl, lwowska szkoła matematyczna) was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the journal '' Studia Mathematica'', founded in 1929. The school was renowned for its productivity and its extensive contributions to subjects such as point-set topology, set theory and functional analysis. The biographies and contributions of these mathematicians were documented in 1980 by their contemporary Kazimierz Kuratowski in his book ''A Half Century of Polish Mathematics: Remembrances and Reflections''. Members Notable members of the Lwów school of mathematics included: * Stefan Banach * Feliks Barański * Władysław Orlicz * Stanisław Saks * Hugo Steinhaus * Stanisław Mazur * Stanisław Ulam * Józef Schreier * Juliusz Schauder * Mark Kac * Antoni Łomnicki * Stefan Kaczmar ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]