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In mathematics, an inner product space or a Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets. Inner products allow the rigorous 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. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.
Kernel may refer to:
In mathematics, positive semidefinite may refer to:
Antisymmetric or skew-symmetric may refer to:
In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.
In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,
Bilinear may refer to:
In mathematics, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions and in the RKHS are close in norm, i.e., is small, then and are also pointwise close, i.e., is small for all . The reverse does not need to be true.
In mathematics, specifically linear algebra, a degenerate bilinear formf(x, y) on a vector space V is a bilinear form such that the map from V to V∗ given by v ↦ is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
In mathematics, in the area of numerical analysis, Galerkin methods convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions.
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over pairs of data points in raw representation.
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described as the composition of at most n reflections.
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz have a two-variable theory that includes all reasonable bilinear forms on the space of test functions. The space itself consists of smooth functions of compact support.
In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:
- the real linear extension αR:Cg × Cg→R of α satisfies αR(iv, iw)=αR(v, w) for all in Cg × Cg;
- the associated hermitian form H(v, w)=αR(iv, w) + iαR(v, w) is positive-definite.
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas.
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign for every nonzero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite.
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular:
References
- Fasshauer, Gregory E. (2011), "Positive definite kernels: Past, present and future" (PDF), Dolomites Research Notes on Approximation, 4: 21–63.
- Stewart, James (1976), "Positive definite functions and generalizations, an historical survey", The Rocky Mountain Journal of Mathematics, 6 (3): 409–434, doi: 10.1216/RMJ-1976-6-3-409 , MR 0430674 .
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