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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 induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space.
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit ...
The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function = + is the boundary value of a holomorphic function F(z) in the upper half-plane. [32] Under these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
In the case where the Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying boundary conditions. In mathematical terms, choosing the boundary ...
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication.
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).
The vector space of all continuous antilinear functions on H is called the anti-dual space or complex conjugate dual space of H and is denoted by ¯ ′ (in contrast, the continuous dual space of H is denoted by ′), which we make into a normed space by endowing it with the canonical norm (defined in the same way as the canonical norm on the ...