Definition
Let denote the inner product on an inner product space and let be a nonempty subset of . A correspondence is called cyclically monotone if for every set of points with it holds that [3]
In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. [1] [2]
Let denote the inner product on an inner product space and let be a nonempty subset of . A correspondence is called cyclically monotone if for every set of points with it holds that [3]
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
In mathematics, an inner product space is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimension are widely used 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.
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.
Distributions, also known as Schwartz distributions or generalized functions, 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.
In mathematics, a linear form is a linear map from a vector space to its field of scalars.
In mathematics, the Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity into functions of the conjugate quantity. In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.
Multi-task learning (MTL) is a subfield of machine learning in which multiple learning tasks are solved at the same time, while exploiting commonalities and differences across tasks. This can result in improved learning efficiency and prediction accuracy for the task-specific models, when compared to training the models separately. Early versions of MTL were called "hints".
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate. It allows in particular for a far reaching generalization of Lagrangian duality.
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.
In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in .
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use.
In mathematics, Hilbert spaces allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space.
In mathematics, a dual system, dual pair, or duality over a field is a triple consisting of two vector spaces and over and a non-degenerate bilinear map . Duality theory, the study of dual systems, is part of functional analysis.
This is a glossary for the terminology in a mathematical field of functional analysis.
In computable analysis, Weihrauch reducibility is a notion of reducibility between multi-valued functions on represented spaces that roughly captures the uniform computational strength of computational problems. It was originally introduced by Klaus Weihrauch in an unpublished 1992 technical report.
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