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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

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David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust

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Erhard Schmidt
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Erhard Schmidt was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu, in the Governorate of Livonia and his advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Functionen nach Systemen vorgeschriebener and was a work on integral equations, together with David Hilbert he made important contributions to functional analysis. After the war, in 1948, Schmidt founded and became the first editor-in-chief of the journal Mathematische Nachrichten, such a war would have cost us half a million young men. But everybody would have admired our victorious leader, now, Hitler has sacrificed half a million Jews and has achieved great things for Germany. I hope some day you will be recompensed but I am still grateful to Hitler, gram–Schmidt process Hilbert–Schmidt operator Lyapunov–Schmidt reduction Schmidt decomposition Hilbert–Schmidt integral operator List of Baltic German scientists Zermelo, Ernst. Beweis, daß jede Menge wohlgeordnet werden kann, reprinted in English translation as Proof that every set can be well-ordered, van Heijenoort 1976, pp. 139–141

4.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, 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 —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist

5.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space