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.
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

8.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

9.
Norm (mathematics)
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A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors. A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the definition below. A simple example is the 2-dimensional Euclidean space R2 equipped with the Euclidean norm, elements in this vector space are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin. The Euclidean norm assigns to each vector the length of its arrow, because of this, the Euclidean norm is often known as the magnitude. A vector space on which a norm is defined is called a vector space. Similarly, a space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a vector space in more than one way. If p =0 then v is the zero vector, by the first axiom, absolute homogeneity, we have p =0 and p = p, so that by the triangle inequality p ≥0. A seminorm on V is a p, V → R with the properties 1. and 2. Every vector space V with seminorm p induces a normed space V/W, called the quotient space, the induced norm on V/W is clearly well-defined and is given by, p = p. A topological vector space is called if the topology of the space can be induced by a norm. If a norm p, V → R is given on a vector space V then the norm of a vector v ∈ V is usually denoted by enclosing it within double vertical lines, such notation is also sometimes used if p is only a seminorm. For the length of a vector in Euclidean space, the notation | v | with single vertical lines is also widespread, in Unicode, the codepoint of the double vertical line character ‖ is U+2016. The double vertical line should not be confused with the parallel to symbol and this is usually not a problem because the former is used in parenthesis-like fashion, whereas the latter is used as an infix operator. The double vertical line used here should not be confused with the symbol used to denote lateral clicks. The single vertical line | is called vertical line in Unicode, the trivial seminorm has p =0 for all x in V. Every linear form f on a vector space defines a seminorm by x → | f |, the absolute value ∥ x ∥ = | x | is a norm on the one-dimensional vector spaces formed by the real or complex numbers. The absolute value norm is a case of the L1 norm

10.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines

11.
Distribution (mathematics)
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Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense, in particular, any locally integrable function has a distributional derivative. The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve differential equations. According to his autobiography, Schwartz introduced the distribution by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles. The basic idea in theory is to reinterpret functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of functions with compact support. Use of the space of smooth, rapidly decreasing test functions gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the sense, but the converse is not true, in general the larger the space of test functions. Distributions are a class of linear functionals that map a set of test functions into the set of real numbers. In the simplest case, the set of test functions considered is D, a distribution T is a linear mapping T, D → R. Instead of writing T, it is conventional to write ⟨ T, φ ⟩ for the value of T acting on a test function φ. A simple example of a distribution is the Dirac delta δ, defined by ⟨ δ, φ ⟩ = φ and its physical interpretation is as the density of a point source. As described next, there are straightforward mappings from both locally integrable functions and Radon measures to corresponding distributions, but not all distributions can be formed in this manner, suppose that f, R → R is a locally integrable function. Then a corresponding distribution Tf may be defined by ⟨ T f, φ ⟩ = ∫ R f φ d x for φ ∈ D and this integral is a real number which depends linearly and continuously on φ. Conversely, the values of the distribution Tf on test functions in D determine the pointwise almost everywhere values of the f on R. In a conventional abuse of notation, f is used to represent both the original function f and the corresponding distribution Tf. This integral also depends linearly and continuously on φ, so that Rμ is a distribution, Distributions may be multiplied by real numbers and added together, so they form a real vector space

12.
Polarization identity
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In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let ∥ x ∥ denote the norm of x and ⟨ x, y ⟩ the inner product of vectors x and y. The various forms given below are all related by the law,2 ∥ u ∥2 +2 ∥ v ∥2 = ∥ u + v ∥2 + ∥ u − v ∥2. The polarization identity can be generalized to other contexts in abstract algebra, linear algebra. If V is a vector space, then the inner product is defined by the polarization identity ⟨ x, y ⟩ =14 ∀ x, y ∈ V. If V is a vector space the inner product is given by the polarization identity, ⟨ x, y ⟩ =14 ∀ x, y ∈ V. Note that this defines a product which is linear in its first. To adjust for contrary definition, one needs to take the complex conjugate, a special case is an inner product given by the dot product, the so-called standard or Euclidean inner product. In this case, common forms of the identity include, u ⋅ v =12, u ⋅ v =12, u ⋅ v =14. The second form of the identity can be written as ∥ u − v ∥2 = ∥ u ∥2 + ∥ v ∥2 −2. This is essentially a form of the law of cosines for the triangle formed by the vectors u, v. In particular, u ⋅ v = ∥ u ∥ ∥ v ∥ cos θ, the basic relation between the norm and the dot product is given by the equation ∥ v ∥2 = v ⋅ v. Forms and of the polarization identity now follow by solving equations for u · v. In linear algebra, the identity applies to any norm on a vector space defined in terms of an inner product by the equation ∥ v ∥ = ⟨ v, v ⟩. This inequality ensures that the magnitude of the above defined cosine ≤1, the choice of the cosine function ensures that when ⟨ u, v ⟩ =0, the angle θ = π/2 or -π/2, where the sign is determined by an orientation on the vector space. In this case, the identities become ⟨ u, v ⟩ =12, ⟨ u, v ⟩ =12, ⟨ u, v ⟩ =14. Conversely, if a norm on a space satisfies the parallelogram law. In functional analysis, introduction of an inner product norm like this often is used to make a Banach space into a Hilbert space, the polarization identities are not restricted to inner products