1.
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

2.
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

3.
Banach space
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In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector 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, Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces, the vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. All norms on a vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space, if X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T, X → Y is denoted by B. In infinite-dimensional spaces, not all maps are continuous. For Y a Banach space, the space B is a Banach space with respect to this norm, if X is a Banach space, the space B = B forms a unital Banach algebra, the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T, X → Y such that T, if one of the two spaces X or Y is complete then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, the Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ. Every normed space X can be embedded in a Banach space. More precisely, there is a Banach space Y and an isometric mapping T, X → Y such that T is dense in Y. If Z is another Banach space such that there is an isomorphism from X onto a dense subset of Z. This Banach space Y is the completion of the normed space X, the underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y. The completion of X is often denoted by X ^, the cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ∥ ∥1 = ∥ x ∥ + ∥ y ∥, ∥ ∥ ∞ = max. In this sense, the product X × Y is complete if and only if the two factors are complete. If M is a linear subspace of a normed space X, there is a natural norm on the quotient space X / M

4.
Stereotype space
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In functional analysis and related areas of mathematics stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. Each pseudocomplete barreled space X is stereotype, a metrizable locally convex space X is stereotype if and only if X is complete. Each infinite dimensional normed space X considered with the X ⋆ -weak topology is not stereotype, there exist stereotype spaces which are not Mackey spaces. Some simple connections between the properties of a stereotype space X and those of its dual space X ⋆ are expressed in the following list of regularities, the first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith in 1952. Further investigations were conducted by B. S. Brudovskii, W. C, waterhouse, K. Brauner, S. S. Akbarov, and E. T. Shavgulidze. Each locally convex space X can be transformed into a space with the help of the standard operations of pseudocompletion and pseudosaturation defined by the following two propositions. If X is a locally convex space, then its pseudosaturation X △ is stereotype. Dually, if X is a locally convex space, then its pseudocompletion X ▽ is stereotype. For arbitrary locally convex space X the spaces X △ ▽ and X ▽ △ are stereotype and it defines two natural tensor products X ⊛ Y, = Hom ⋆, X ⊙ Y, = Hom. This condition is weaker than the existence of the Schauder basis, the following proposition holds, If two stereotype spaces X and Y have the stereotype approximation property, then the spaces Hom, X ⊛ Y and X ⊙ Y have the stereotype approximation property as well. In particular, if X has the approximation property, then the same is true for X ⋆. This allows to reduce the list of counterexamples in comparison with the Banach theory, the arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras in these theories become Hopf algebras in the algebraic sense. Schaefer, Helmuth H. Topological vector spaces, Robertson, A. P. Robertson, W. J. Topological vector spaces. The Pontrjagin duality theorem in linear spaces, on k- and c-reflexivity of locally convex vector spaces. Brauner, K. Duals of Fréchet spaces and a generalization of the Banach-Dieudonné theorem, Akbarov, S. S. Pontryagin duality in the theory of topological vector spaces and in topological algebra. Akbarov, S. S. Holomorphic functions of exponential type, envelopes and refinements in categories, with applications to functional analysis. On two classes of spaces reflexive in the sense of Pontryagin, Akbarov, S. S. Pontryagin duality and topological algebras

5.
Inner product space
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In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors, inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. An inner product induces a associated norm, thus an inner product space is also a normed vector space. A complete space with a product is called a Hilbert space. An space with a product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C, formally, an inner product space is a vector space V over the field F together with an inner product, i. e. Some authors, especially in physics and matrix algebra, prefer to define the inner product, then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨ x, y ⟩ as ⟨ y | x ⟩, respectively y † x. Here the kets and columns are identified with the vectors of V and this reverse order is now occasionally followed in the more abstract literature, taking ⟨ x, y ⟩ to be conjugate linear in x rather than y. A few instead find a ground by recognizing both ⟨ ⋅, ⋅ ⟩ and ⟨ ⋅ | ⋅ ⟩ as distinct notations differing only in which argument is conjugate linear. There are various reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense, the basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, however in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as used in quantum computation, are automatically metrically complete. In some cases we need to consider non-negative semi-definite sesquilinear forms and this means that ⟨ x, x ⟩ is only required to be non-negative

6.
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

7.
Boundary (topology)
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In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, an element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set, notations used for boundary of a set S include bd, fr, and ∂S. Some authors use the term instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology. However, frontier sometimes refers to a different set, which is the set of points which are not actually in the set. A connected component of the boundary of S is called a component of S. If the set consists of points only, then the set has only a boundary. There are several definitions to the boundary of a subset S of a topological space X. The intersection of the closure of S with the closure of its complement, the set of points p of X such that every neighborhood of p contains at least one point of S and at least one point not of S. Consider the real line R with the usual topology, one has ∂ = ∂ = ∂ = ∂∅ = ∅ ∂Q = R ∂ = These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. In the space of rational numbers with the topology, the boundary of. The boundary of a set is a topological notion and may change if one changes the topology, for example, given the usual topology on R2, the boundary of a closed disk Ω = is the disks surrounding circle, ∂Ω =. If the disk is viewed as a set in R3 with its own usual topology, i. e. Ω =, then the boundary of the disk is the disk itself, ∂Ω = Ω. If the disk is viewed as its own space, then the boundary of the disk is empty. The boundary of a set is closed, the boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set. A set is the boundary of some open set if and only if it is closed, the boundary of a set is the boundary of the complement of the set, ∂S = ∂. The interior of the boundary of a set is the empty set. Hence, p is a point of a set if and only if every neighborhood of p contains at least one point in the set

8.
Affine space
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A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, a property is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin, hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the elements of the space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the vector is called the origin. Adding a fixed vector to the elements of a subspace of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating the linear subspace by the translation vector, in finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that space are the solutions of the corresponding homogeneous linear system. Linear subspaces, in contrast, always contain the origin of the vector space, the dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of one is an affine line. An affine space of dimension 2 is an affine plane, an affine subspace of dimension n –1 in an affine space or a vector space of dimension n is an affine hyperplane. The following characterization may be easier to understand than the formal definition. Imagine that Alice knows that a point is the actual origin. Two vectors, a and b, are to be added, similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a combination is 1, then Alice. If Alice travels to λa + b then Bob can similarly travel to p + λ + = λa + b, under this condition, for all coefficients λ + =1, Alice and Bob describe the same point with the same linear combination, despite using different origins

9.
Bilinear map
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In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Let V, W and X be three vector spaces over the base field F. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. If V = W and we have B = B for all v, w in V, the case where X is the base field F, and we have a bilinear form, is particularly useful. The definition works without any changes if instead of vector spaces over a field F and it generalizes to n-ary functions, where the proper term is multilinear. This satisfies B = r ⋅ B B = B ⋅ s for all m in M, n in N, r in R and s in S, a first immediate consequence of the definition is that B = 0X whenever v = 0V or w = 0W. This may be seen by writing the zero vector 0X as 0 ⋅ 0X and moving the scalar 0 outside, in front of B, the set L of all bilinear maps is a linear subspace of the space of all maps from V × W into X. If V, W, X are finite-dimensional, then so is L, for X = F, i. e. bilinear forms, the dimension of this space is dim V × dim W. To see this, choose a basis for V and W, then each bilinear map can be represented by the matrix B. Now, if X is a space of dimension, we obviously have dim L = dim V × dim W × dim X. Matrix multiplication is a bilinear map M × M → M. If a vector space V over the real numbers R carries an inner product, in general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F. If V is a space with dual space V∗, then the application operator. Let V and W be vector spaces over the base field F. If f is a member of V∗ and g a member of W∗, the cross product in R3 is a bilinear map R3 × R3 → R3. Let B, V × W → X be a bilinear map, and L, U → W be a linear map, then ↦ B is a bilinear map on V × U

10.
Unit sphere
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Usually a specific point has been distinguished as the origin of the space under study and it is understood that a unit sphere or unit ball is centered at that point. Therefore one speaks of the ball or the unit sphere. For example, a sphere is the surface of what is commonly called a circle, while such a circles interior. Similarly, a sphere is the surface of the Euclidean solid known colloquially as a sphere, while the interior. A unit sphere is simply a sphere of radius one, the importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere. In Euclidean space of n dimensions, the sphere is the set of all points which satisfy the equation x 12 + x 22 + ⋯ + x n 2 =1. The volume of the ball in n dimensions, which we denote Vn. It is V n = π n /2 Γ = { π n /2 /, I f n ≥0 i s e v e n, π ⌊ n /2 ⌋2 ⌈ n /2 ⌉ / n. I f n ≥0 i s o d d, where n. is the double factorial, the surface areas and the volumes for some values of n are as follows, where the decimal expanded values for n ≥2 are rounded to the displayed precision. The An values satisfy the recursion, A0 =0 A1 =2 A2 =2 π A n =2 π n −2 A n −2 for n >2. The Vn values satisfy the recursion, V0 =1 V1 =2 V n =2 π n V n −2 for n >1. The surface area of a sphere with radius r is An rn−1. For instance, the area is A = 4π r 2 for the surface of the ball of radius r. The volume is V = 4π r 3 /3 for the ball of radius r. More precisely, the unit ball in a normed vector space V. It is the interior of the unit ball of. The latter is the disjoint union of the former and their common border, the shape of the unit ball is entirely dependent on the chosen norm, it may well have corners, and for example may look like n, in the case of the norm l∞ in Rn