Support (mathematics)
is the subset of the function domain of elements that are not mapped to zero.is instead defined as the smallest closed set containing all points not mapped to zero.The notion of support also extends in a natural way to functions taking values in more general sets thanis the intersection of all closed sets that contain the set-theoretic support ofThe notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require thatIn good cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example.As an intuition for more complex examples, and in the language of limits, for anythat vanishes at infinity can be approximated by choosing an appropriate compact subsetequipped with Lebesgue measure), then one typically identifies functions that are equalis an arbitrary set containing zero, the concept of support is immediately generalizable to functionsSupport may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero.is the countable set of all integer sequences that have only finitely many nonzero entries.[7] In probability theory, the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution.There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra, rather than on a topological space.[8] Note that the word support can refer to the logarithm of the likelihood of a probability density function.[9] It is possible also to talk about the support of a distribution, such as the Dirac delta functionwhich are smooth functions with support not including the pointis an open set in Euclidean space such that, for all test functionsIn Fourier analysis in particular, it is interesting to study the singular support of a distribution.This has the intuitive interpretation as the set of points at which a distribution fails to be a smooth function.For example, the Fourier transform of the Heaviside step function can, up to constant factors, be considered to beis clearly a special point, it is more precise to say that the transform of the distribution has singular supportIt can be expressed as an application of a Cauchy principal value improper integral.For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis.Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).An abstract notion of family of supports on a topological spacesuitable for sheaf theory, was defined by Henri Cartan.In extending Poincaré duality to manifolds that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example Alexander–Spanier cohomology.Bredon, Sheaf Theory (2nd edition, 1997) gives these definitions.is a family of supports, if it is down-closed and closed under finite union.