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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 first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
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 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
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.
Let be an arbitrary set and a Hilbert space of real-valued functions on , equipped with pointwise addition and pointwise scalar multiplication.The evaluation functional over the Hilbert space of functions is a linear functional that evaluates each function at a point ,
A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map i is continuous. Identifying H with its dual space H * , the adjoint to i is the map i ∗ : H = H ∗ → Φ ∗ . {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
The Hilbert Curve is commonly used among rendering images or videos. Common programs such as Blender and Cinema 4D use the Hilbert Curve to trace the objects, and render the scene. [citation needed] The slicer software used to convert 3D models into toolpaths for a 3D printer typically has the Hilbert curve as an option for an infill pattern.